View source for Banach-Mazur compactum

''geometry of the''

The Banach–Mazur compactum provides the natural framework for a quantitative study of finite-dimensional normed spaces with particular emphasis on the role of the [[Dimension|dimension]]. Since every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b1101001.png" />-dimensional normed real vector space is easily seen to be isometric to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b1101002.png" /> equipped with a suitable [[Norm|norm]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b1101003.png" />, the typical normed space discussed here will be <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b1101004.png" />. The unit ball <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b1101005.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b1101006.png" /> is a symmetric [[Convex body|convex body]]. Conversely, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b1101007.png" /> is a symmetric convex body in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b1101008.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b1101009.png" /> induces on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b11010010.png" /> the norm <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b11010011.png" /> and the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b11010012.png" /> has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b11010013.png" /> as its unit ball. Thus, the study of finite-dimensional normed spaces is parallel to that of symmetric convex bodies. If one assumes that an [[Inner product|inner product]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b11010014.png" /> defines the Euclidean structure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b11010015.png" />, then the dual space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b11010016.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b11010017.png" /> (cf. also [[Adjoint space|Adjoint space]]) has its norm given by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b11010018.png" />. The unit ball <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b11010019.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b11010020.png" /> is the polar body of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b11010021.png" />.

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b11010022.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b11010023.png" /> be two <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b11010024.png" />-dimensional normed spaces. It is well-known that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b11010025.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b11010026.png" /> are isomorphic (cf. [[Isomorphism|Isomorphism]]). The Banach–Mazur distance

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b11010027.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b11010028.png" /></td> </tr></table>

measures how well-isomorphic the two spaces are. An equivalent geometric interpretation is that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b11010029.png" /> is the smallest possible <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b11010030.png" /> for which there exists a linear invertible transformation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b11010031.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b11010032.png" />. It is easy to check that for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b11010033.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b11010034.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b11010035.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b11010036.png" /> is symmetric); <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b11010037.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b11010038.png" /> is isometric to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b11010039.png" />; and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b11010040.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b11010041.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b11010042.png" /> are the dual spaces of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b11010043.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b11010044.png" />, respectively.

Consider the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b11010045.png" /> of all equivalence classes of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b11010046.png" />-dimensional normed spaces, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b11010047.png" /> is equivalent to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b11010048.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b11010049.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b11010050.png" /> are isometric. Then it is not hard to check that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b11010051.png" /> becomes a contractible compact [[Metric space|metric space]] with [[Metric|metric]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b11010052.png" />: the triangle inequality is a consequence of

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b11010053.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>

which can be easily verified for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b11010054.png" />. The metric space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b11010055.png" /> is usually called the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b11010056.png" />-dimensional Banach–Mazur compactum (or Minkowski compactum). In the sequel, instead of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b11010057.png" />, the  "multiplicative"  distance <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b11010058.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b11010059.png" /> is considered.

The structure of the Banach–Mazur compactum is still to be explored. There are many interesting quantitative questions one may ask and, besides a few fundamental results, most of these questions remain open. Below, some basic problems and final or partial answers to them are given. Several other important topics are not mentioned here: the interested reader is referred to the books [[#References|[a39]]], [[#References|[a43]]], [[#References|[a44]]], [[#References|[a58]]], and to the survey papers [[#References|[a21]]], [[#References|[a29]]], [[#References|[a30]]], [[#References|[a38]]], [[#References|[a41]]], and [[#References|[a53]]].

==The diameter of the compactum.==
The first classical result on the geometry of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b11010060.png" /> is John's theorem [[#References|[a26]]] on the maximal possible distance to the Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b11010061.png" />. Starting with an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b11010062.png" />-dimensional space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b11010063.png" />, F. John considered the linear transformation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b11010064.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b11010065.png" /> for which the Euclidean unit ball <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b11010066.png" /> is the ellipsoid of minimal volume containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b11010067.png" />. Using a variational argument, he obtained the following precise information about <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b11010068.png" />:

i) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b11010069.png" />;

ii) there exist <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b11010070.png" /> contact points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b11010071.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b11010072.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b11010073.png" /> (i.e., <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b11010074.png" />), and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b11010075.png" /> such that the following representation of the identity holds: for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b11010076.png" />,

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In particular, i) gives an upper bound for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b11010078.png" /> (John's theorem): <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b11010079.png" /> for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b11010080.png" />.

This estimate is sharp: one can see that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b11010081.png" />. John's theorem and a direct application of the multiplicative triangle inequality (a1) give some first information on the diameter of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b11010082.png" />:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b11010083.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a2)</td></tr></table>

A most natural question arising from (a2) is to determine <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b11010084.png" />. The exact value of this quantity is known only when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b11010085.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b11010086.png" />, by results of E. Asplund [[#References|[a1]]] and W. Stromquist [[#References|[a48]]]. Even the question of determining up to constants the order of growth of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b11010087.png" /> as a function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b11010088.png" /> remained open for a long time. The answer was given by E.D. Gluskin [[#References|[a18]]] in 1981 (Gluskin's theorem): There exists an absolute constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b11010089.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b11010090.png" /> for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b11010091.png" />.

Therefore, the diameter of the Banach–Mazur compactum is of the order of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b11010092.png" />. In his solution of the problem, Gluskin does not describe the pair of spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b11010093.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b11010094.png" /> explicitly (in fact, no explicit example of two spaces with distance considerably larger than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b11010095.png" /> is known (1996)).

The existence of a pair with distance of the order of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b11010096.png" /> has been established by probabilistic arguments. Gluskin [[#References|[a21]]] describes his idea as follows: Consider the pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b11010097.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b11010098.png" /> be a random matrix whose entries are independent standard Gaussian variables. It is not hard to see that the inequality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b11010099.png" /> holds with high probability for matrices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100100.png" /> of this form. This suggests that by  "spoiling"  the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100101.png" /> a bit it is possible to obtain spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100102.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100103.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100104.png" /> for every linear isomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100105.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100106.png" />. This is actually the case.

Gluskin considered spaces whose unit ball is a symmetric convex body of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100107.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100108.png" /> is the standard orthonormal basis of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100109.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100110.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100111.png" />, are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100112.png" /> independent random points chosen from the Euclidean unit sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100113.png" /> equipped with the rotationally invariant probability measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100114.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100115.png" /> is a pair of independent  "Gluskin spaces" , one can show that the probability

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100116.png" /></td> </tr></table>

is exponentially small, uniformly in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100117.png" />. Combining this with exact estimates on the cardinality of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100118.png" />-nets in suitable spaces of operators one gets that, for most pairs <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100119.png" />, the inequality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100120.png" /> holds true on a sufficiently fine net in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100121.png" />. A standard argument, depending, however, on the precise estimates above, allows one to pass from the net to an arbitrary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100122.png" />, thus showing that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100123.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100124.png" />.

Besides settling the question on the diameter of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100125.png" />, Gluskin's method of considering random spaces proved to be extremely influential and fruitful. Random spaces provided the right framework for establishing  "pathological behaviour"  and solving several other major open problems in the asymptotic theory of finite-dimensional normed spaces. An example of this is Szarek's finite-dimensional analogue of Enflo's example of a space without the approximation property [[#References|[a12]]]: There exist <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100126.png" />-dimensional normed spaces whose basis constant is of the order of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100127.png" /> [[#References|[a51]]]. See also [[#References|[a19]]], [[#References|[a32]]] and subsequent work of S.J. Szarek and P. Mankiewicz, in which Gluskin spaces play a central role.

==Extremal problems for special families of spaces.==
In many important cases, the Banach–Mazur distance <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100128.png" /> is significantly smaller than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100129.png" />. For example, when both <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100130.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100131.png" /> are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100132.png" />-spaces (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100133.png" />) one has the following classical Gurarii–Kadec–Macaev estimates [[#References|[a25]]]:

i) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100134.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100135.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100136.png" />;

ii) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100137.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100138.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100139.png" /> are absolute constants and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100140.png" />.

Much work has been done in the direction of estimating <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100141.png" /> for some important families <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100142.png" />. One of the methods invented for dealing with such problems is the method of random orthogonal factorization, which has its origin in work of N. Tomczak-Jaegermann [[#References|[a55]]] and Y. Benyamini and Y. Gordon [[#References|[a3]]]. The idea is to use the average of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100143.png" /> with respect to the probability [[Haar measure|Haar measure]] on the orthogonal group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100144.png" /> as an upper bound for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100145.png" />. Technically, this process depends upon passing from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100146.png" /> to matrices whose entries are independent standard Gaussian variables and then use the theory of Gaussian processes to estimate this average.

Using this method, one can prove a general inequality involving the type-2 constant of the spaces [[#References|[a3]]], [[#References|[a8]]]:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100147.png" /></td> </tr></table>

for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100148.png" />. This estimate may be viewed as a generalization of the estimates for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100149.png" /> given above. This was further improved by J. Bourgain and V.D. Milman [[#References|[a5]]] to

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100150.png" /></td> </tr></table>

See [[#References|[a39]]] or [[#References|[a58]]] for definitions and theory of the type and co-type parameters.

Two more results obtained by the same method and having an obvious geometric flavour are as follows.

Suppose that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100151.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100152.png" /> are two <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100153.png" />-dimensional spaces such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100154.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100155.png" /> have few extreme points, in the sense that their cardinality is bounded by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100156.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100157.png" /> is a fixed positive number. Then

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100158.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100159.png" /> is a constant depending only on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100160.png" /> (see [[#References|[a8]]]).

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100161.png" /> be an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100162.png" />-dimensional space. It has been shown [[#References|[a5]]] that the distance from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100163.png" /> to its dual <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100164.png" /> is  "small" . More precisely, for some constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100165.png" />,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100166.png" /></td> </tr></table>

The best-possible exponent of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100167.png" /> is not yet known (1996).

Both these results indicate that the distance between two spaces whose unit balls are  "quite different"  as convex bodies, should not be of the order of the diameter of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100168.png" />. Recall that in Gluskin's theorem the spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100169.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100170.png" /> with distance <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100171.png" /> were of the same nature ( "spoiled"  <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100172.png" />-spaces).

Now, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100173.png" /> to be the family of all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100175.png" />-symmetric spaces. Such a space possesses a basis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100176.png" /> with the property that for every choice of scalars <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100177.png" />, every choice of signs <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100178.png" />, and every permutation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100179.png" /> of the set of indices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100180.png" />,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100181.png" /></td> </tr></table>

Gluskin [[#References|[a20]]] and Tomczak-Jaegermann have proved that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100182.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100183.png" /> is bounded by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100184.png" /> up to some factor logarithmic in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100185.png" />. Finally, Tomczak-Jaegermann [[#References|[a56]]] succeeded to remove this extra factor, thus answering completely the following question on the maximal distance between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100186.png" />-symmetric spaces (Tomczak-Jaegermann theorem): For two <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100187.png" />-symmetric spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100188.png" /> one has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100189.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100190.png" /> is an absolute constant.

Note that every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100191.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100192.png" />-symmetric space, therefore the estimate above is clearly optimal (consider the distance between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100193.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100194.png" />).

Another important family of spaces in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100195.png" /> is the family of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100197.png" />-unconditional spaces. These are spaces for which there exists a basis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100198.png" /> with the property that for every choice of scalars <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100199.png" /> and every choice of signs <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100200.png" />,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100201.png" /></td> </tr></table>

One naturally asks for the order of

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100202.png" /></td> </tr></table>

The final answer to this problem is not known (1996): it is conjectured that the right order is close to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100203.png" />. J. Lindenstrauss and A. Szankowski [[#References|[a31]]] have shown that it has to be smaller than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100204.png" />: There exists a constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100205.png" /> such that whenever <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100206.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100207.png" /> are two <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100208.png" />-unconditional spaces in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100209.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100210.png" /> for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100211.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100212.png" /> is a constant depending only on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100213.png" />. The constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100214.png" /> obtained in [[#References|[a31]]] is given by a complicated expression. One knows that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100215.png" /> and numerical calculations show that it should be <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100216.png" />.

==Radius of the compactum with respect to a fixed centre.==
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100217.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100218.png" /> be the radius of the Banach–Mazur compactum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100219.png" /> with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100220.png" />, defined by

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100221.png" /></td> </tr></table>

In this terminology, John's theorem states that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100222.png" />. It is natural to ask about sharp analogues of this result if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100223.png" /> is replaced by other standard <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100224.png" />-dimensional spaces. In this direction, A. Pelczynski [[#References|[a41]]] asked a question of obvious geometric importance: What is the order of the maximal possible  "Banach–Mazur distance to the cube"  <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100225.png" />. Until recently, the only information available was that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100226.png" />, a consequence of John's theorem. It turns out that none of these two estimates is sharp: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100227.png" /> ([[#References|[a7]]]; the Bourgain–Szarek estimate). On the way to this estimate, very interesting information on the relation between a symmetric convex body <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100228.png" /> and its minimal volume ellipsoid has been obtained. Assuming that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100229.png" /> is the minimal ellipsoid of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100230.png" />, as in John's argument, it was proved that for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100231.png" /> one can choose <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100232.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100233.png" />, among the contact points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100234.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100235.png" />, such that for every choice of scalars <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100236.png" />,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100237.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a3)</td></tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100238.png" /></td> </tr></table>

The important part in this string of inequalities is, of course, the first one. This is a strengthened version of the classical Dvoretzky–Rogers lemma [[#References|[a11]]], which implied a similar inequality only for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100239.png" />. It can also be stated in the form of a  "proportional factorization result"  (proportional Dvoretzky–Rogers factorization, [[#References|[a7]]]): Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100240.png" /> be an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100241.png" />-dimensional space. For every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100242.png" /> one can find an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100243.png" /> and two operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100244.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100245.png" />, such that the identity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100246.png" /> is written as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100247.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100248.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100249.png" /> is a function depending only on the proportion <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100250.png" />.

Using this result, Bourgain and Szarek gave a final answer to the problem of the uniqueness, up to constant, of the centre of the Banach–Mazur compactum. This can be made a precise question as follows: Does there exist a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100251.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100252.png" />, such that for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100253.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100254.png" /> one must have <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100255.png" />? In other words, are all the  "asymptotic centres"  of the Banach–Mazur compactum close to the Euclidean space? The answer is negative and the main tool in the proof is the proportional Dvoretzky–Rogers result (non-uniqueness of an asymptotic centre for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100257.png" />; [[#References|[a7]]]): Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100258.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100259.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100260.png" /> for some absolute constant, but <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100261.png" />. Therefore, there exist asymptotic centres of the Banach–Mazur compactum with distance to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100262.png" /> of the order of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100263.png" />.

The same inequality allowed Bourgain and Szarek to show that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100264.png" />. It is now known [[#References|[a54]]], [[#References|[a17]]] that (a3) holds true with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100265.png" />. This gives a better upper bound for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100266.png" />, which, however, does not seem to give the right order of magnitude (Banach–Mazur distance to the cube; [[#References|[a52]]], [[#References|[a16]]]): There exist two absolute constants <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100267.png" /> such that

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100268.png" /></td> </tr></table>

The lower bound <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100269.png" />, due to Szarek, shows that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100270.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100271.png" /> are not asymptotic centres of the Banach–Mazur compactum (these are actually essentially the only explicit examples of spaces for which this property has been established). In Szarek's work, the space with distance to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100272.png" /> of the order of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100273.png" /> is once again a Gluskin space. So, the problem of the distance to the cube remains open (even without a strong conjecture about what the order of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100274.png" /> should be, 1996). The question of the best possible exponent of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100275.png" /> in the proportional Dvoretzky–Rogers factorization is also open (1996): by [[#References|[a17]]] and [[#References|[a46]]] it must lie between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100276.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100277.png" />.

==Sections and projections of a symmetric convex body: distance to Euclidean space.==
Some fundamental results on the local structure (i.e., the structure of subspaces and quotients) of finite-dimensional normed spaces are closely related to the Banach–Mazur compactum. The starting point of the asymptotic theory of finite-dimensional normed spaces is Dvoretzky's theorem [[#References|[a9]]], [[#References|[a10]]] on almost spherical sections of symmetric convex bodies: For every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100278.png" /> and every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100279.png" />-dimensional normed space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100280.png" />, there exist an integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100281.png" /> and a subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100282.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100283.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100284.png" />, such that

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100285.png" /></td> </tr></table>

In other words, every symmetric convex body in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100286.png" /> has sections which are almost ellipsoids and whose dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100287.png" /> can be chosen to increase to infinity as the dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100288.png" /> of the original body tends to infinity. Being probabilistic in nature, the proof actually gives that most of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100289.png" />-dimensional subspaces of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100290.png" /> are almost Euclidean. The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100291.png" /> dependence on the dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100292.png" /> (first obtained by Milman in [[#References|[a34]]]) is exact, as one can see by considering <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100293.png" />. There are several other proofs of Dvoretzky's theorem: see [[#References|[a13]]], [[#References|[a49]]]. The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100294.png" /> dependence on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100295.png" /> (established in [[#References|[a22]]], [[#References|[a47]]]) does not seem to be the right one: see [[#References|[a37]]] for related remarks and conjectures.

A rough description of Milman's argument, which was further exploited in [[#References|[a15]]], runs as follows: consider the quantity

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100296.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100297.png" /> is the rotationally invariant probability measure on the Euclidean unit sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100298.png" />. Without loss of generality one may assume that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100299.png" /> is the ellipsoid of maximal volume contained in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100300.png" />. Then, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100301.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100302.png" />-Lipschitz function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100303.png" /> and the isoperimetric inequality on the sphere implies that the values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100304.png" /> are highly concentrated around its expectation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100305.png" /> (cf. also [[Isoperimetric inequality|Isoperimetric inequality]]). One can extract subspaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100306.png" /> of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100307.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100308.png" /> up to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100309.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100310.png" />. The second main tool is the Dvoretzky–Rogers lemma (in a form dual to (a3)). A simple computation based on the information given by (a3) shows that, under the above hypotheses, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100311.png" /> has to be at least <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100312.png" />.

In many cases one has an estimate for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100313.png" /> which allows almost Euclidean subspaces of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100314.png" /> of dimension even proportional to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100315.png" />. See [[#References|[a15]]], where this is verified for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100316.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100317.png" />. A different approach, which has its origin in [[#References|[a27]]] and was further simplified and generalized in [[#References|[a50]]], shows that this is a common property of all spaces whose unit ball <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100318.png" /> has  "small"  volume ratio (theorem on volume ratio and Euclidean sections): Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100319.png" /> be an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100320.png" />-dimensional normed space and set

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100321.png" /></td> </tr></table>

where the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100322.png" /> is taken over all ellipsoids <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100323.png" /> contained in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100324.png" />. For every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100325.png" /> there exist subspaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100326.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100327.png" />, such that

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100328.png" /></td> </tr></table>

Note that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100329.png" /> is bounded by a constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100330.png" />, then this theorem gives subspaces of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100331.png" /> of any proportional dimension and isomorphic to the Euclidean space up to a constant depending only on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100332.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100333.png" /> (independent of the dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100334.png" />). A simple computation shows that the volume ratio of all spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100335.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100336.png" />, is uniformly bounded.

A crucial inequality in the same direction of a  "proportional theory"  of finite-dimensional spaces is Milman's low <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100338.png" />-estimate, [[#References|[a35]]], which holds for every space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100339.png" />: There exists a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100340.png" /> such that for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100341.png" /> and every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100342.png" /> one can find a subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100343.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100344.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100345.png" /> and satisfying

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100346.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a4)</td></tr></table>

for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100347.png" />.

Again, the proof is probabilistic in nature and shows that this estimate holds true for most subspaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100348.png" /> of a given dimension. There are several proofs of the Milman inequality; see [[#References|[a40]]] and [[#References|[a23]]] for the best possible dependence on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100349.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100350.png" />. Note that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100351.png" /> is half the mean width of the unit ball of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100352.png" /> and that (a4) is equivalent to

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100353.png" /></td> </tr></table>

Thus, the geometric interpretation of Milman's inequality is that the diameter of the proportional sections of a symmetric convex body is controlled by the mean width of the body up to a function depending only on the proportion <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100354.png" />.

An important consequence of Milman's inequality is Milman's quotient-of-subspace theorem, [[#References|[a35]]], which states that, starting with any space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100355.png" /> and any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100356.png" />, one can find a quotient of a subspace of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100357.png" /> with dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100358.png" /> and which is Euclidean up to a constant depending only on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100359.png" />. This should be compared with the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100360.png" />-dimension of the Euclidean subspaces in Dvoretzky's theorem and the fact that the corresponding dimension for Euclidean quotients does not exceed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100361.png" /> in general. More precisely (the Milman quotient-of-subspace theorem): Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100362.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100363.png" />. There exist subspaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100364.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100365.png" /> and

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100366.png" /></td> </tr></table>

Crucial results from [[#References|[a14]]], [[#References|[a28]]] and [[#References|[a42]]] always allow one to have <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100367.png" /> logarithmic in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100368.png" /> for a suitable choice of the Euclidean structure in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100369.png" />. Then, an iteration scheme based on double applications of Milman's inequality completes the proof.

The quotient-of-subspace theorem has found several applications even in the context of classical convexity, such as the inverse Santaló inequality and the inverse Brunn–Minkowski inequality.

It was proved by W. Blaschke for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100370.png" />, and by L. Santaló for general <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100371.png" />, that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100372.png" /> for every symmetric convex body <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100373.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100374.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100375.png" /> is the polar body of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100376.png" />. The quotient-of-subspace theorem enabled Bourgain and Milman [[#References|[a6]]] to prove that

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100377.png" /></td> </tr></table>

for every body <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100378.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100379.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100380.png" /> are positive absolute constants. This shows that the affinely invariant volume product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100381.png" /> is the same for all symmetric convex bodies, up to an absolute constant.

The classical Brunn–Minkowski inequality (cf. also [[Brunn–Minkowski theorem|Brunn–Minkowski theorem]]) states that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100382.png" /> for every pair of convex bodies in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100383.png" />. It is easy to check that, in general, no inverse to this inequality can hold. However, Milman [[#References|[a36]]] proved the following: There exists a mapping sending each symmetric convex body <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100384.png" /> to some linear transformation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100385.png" /> such that for every pair of symmetric convex bodies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100386.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100387.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100388.png" /> the following inequality holds:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100389.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100390.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100391.png" /> is an absolute constant. The body <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100392.png" /> is called an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100394.png" />-position of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100395.png" />.

An example of the interaction between the local structure of normed spaces and the geometry of the Banach–Mazur compactum is the isomorphic finite-dimensional version of the homogeneous space problem. A question raised by S. Banach in [[#References|[a2]]] is whether an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100396.png" />-dimensional normed space all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100397.png" />-dimensional subspaces of which are pairwise isometric for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100398.png" />, must be isometric to the Euclidean space. In most cases this has been answered in the affirmative by M. Gromov [[#References|[a24]]].

An isomorphic version of the same problem was studied by Bourgain [[#References|[a4]]]. He showed that there are a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100399.png" /> and a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100400.png" /> such that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100401.png" /> is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100402.png" />-dimensional space with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100403.png" /> for every pair of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100404.png" />-dimensional subspaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100405.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100406.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100407.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100408.png" />.

Mankiewicz and Tomczak–Jaegermann completed this result. They proved in [[#References|[a33]]] that there is a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100409.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100410.png" /> for every space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100411.png" /> all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100412.png" />-dimensional subspaces of which are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100413.png" />-isomorphic. The positive answer to the question involves most of the basic methods and results mentioned in the previous sections.

==Weak Banach–Mazur distance.==
Tomczak–Jaegermann [[#References|[a57]]] has introduced a different way of measuring the distance between two <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100414.png" />-dimensional normed spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100415.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100416.png" />. She first considered the weak factorization norm of the identity

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100417.png" /></td> </tr></table>

where the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100418.png" /> is taken over all measure spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100419.png" /> and all mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100420.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100421.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100422.png" />. Then, one symmetrizes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100423.png" /> to define the weak distance between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100424.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100425.png" /> as

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100426.png" /></td> </tr></table>

One can easily check that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100427.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100428.png" />. On the other hand, one can see that, with high probability, the weak distance between two Gluskin spaces is bounded by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100429.png" />. So, the weak distance may differ much from the Banach–Mazur distance. Actually, M. Rudelson [[#References|[a45]]] has given an example of a pair of spaces with weak distance less than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100430.png" /> and Banach–Mazur distance of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100431.png" />.

The diameter of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100432.png" /> with respect to the weak distance is much smaller than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100433.png" />. Rudelson [[#References|[a45]]] has proved that

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100434.png" /></td> </tr></table>

for every pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100435.png" />. It has been conjectured that the weak distance in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100436.png" /> is always bounded by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110100/b110100437.png" />.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E. Asplund,   "Comparison between plane symmetric convex bodies and parallelograms"  ''Math. Scand.'' , '''8'''  (1960)  pp. 171–180</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  S. Banach,   "Théorie des opérations linéaires" , PWN  (1932)  (New Edition: 1979)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  Y. Benyamini,   Y. Gordon,   "Random factorization of operators between Banach spaces"  ''J. d'Anal. Math.'' , '''39'''  (1981)  pp. 45–74</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  J. Bourgain,   "On finite dimensional homogeneous Banach spaces" , ''Lecture Notes in Mathematics'' , '''1317''' , Springer  (1988)  pp. 232–238</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  J. Bourgain,   V.D. Milman,   "Distances between normed spaces, their subspaces and quotient spaces"  ''Integral Eq. 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