View source for Mixed-volume theory

A branch of the theory of convex bodies concerned with the functionals that arise in the study of linear combinations of bodies (see [[Addition of sets|Addition of sets]]).

The volume <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064260/m0642601.png" /> of a linear combination <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064260/m0642602.png" /> of convex bodies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064260/m0642603.png" /> in a Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064260/m0642604.png" /> with coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064260/m0642605.png" /> is a homogeneous polynomial of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064260/m0642606.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064260/m0642607.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/m/m064/m064260/m0642608.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>

The coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064260/m0642609.png" /> are assumed to be symmetric with respect to permutations of the subscripts and are denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064260/m06426010.png" />, since they depend only on the bodies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064260/m06426011.png" />. These coefficients are called the mixed volumes of the bodies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064260/m06426012.png" />.

The significance of this theory lies in the universality of the concept of mixed volumes: by substituting concrete bodies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064260/m06426013.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064260/m06426014.png" />, one can obtain various quantities related to a body <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064260/m06426015.png" />. These include: its volume, its surface area, the surface integral of the [[elementary symmetric function]] of its principal curvatures (in the case of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064260/m06426016.png" />-smooth body), and also the corresponding characteristics of its projections to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064260/m06426017.png" />-dimensional planes, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064260/m06426018.png" />. A special case of the expression (*) is the Steiner formula for volumes of parallel bodies in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064260/m06426019.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/m/m064/m064260/m06426020.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064260/m06426021.png" /> is the volume, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064260/m06426022.png" /> the surface area, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064260/m06426023.png" /> the total mean curvature of the original body, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064260/m06426024.png" /> the volume of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064260/m06426025.png" />-neighbourhood of it. The mixed volume <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064260/m06426026.png" /> is invariant under parallel translation of any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064260/m06426027.png" />, is monotonic (with respect to inclusion of bodies), continuous and non-negative; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064260/m06426028.png" /> if and only if it is possible to choose a segment in every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064260/m06426029.png" /> such that these segments are linearly independent (see [[#References|[1]]]).

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064260/m06426030.png" /> is the projection of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064260/m06426031.png" /> to a hypersurface orthogonal to a segment <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064260/m06426032.png" /> of length one, 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/m/m064/m064260/m06426033.png" /></td> </tr></table>

The volume of the projection of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064260/m06426034.png" /> to a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064260/m06426035.png" />-dimensional subspace is called its <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064260/m06426037.png" />-th cross-sectional measure (or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064260/m06426039.png" />-quermass). Establishing relations between the mean values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064260/m06426040.png" /> of these measures is one of the concerns of [[Integral geometry|integral geometry]]. Up to a multiplicative constant, the functionals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064260/m06426041.png" /> coincide with the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064260/m06426043.png" />-th integral curvatures:

<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/m/m064/m064260/m06426044.png" /></td> </tr></table>

(<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064260/m06426045.png" /> occurrences of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064260/m06426046.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064260/m06426047.png" /> occurrences of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064260/m06426048.png" />), where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064260/m06426049.png" /> is the unit sphere. For a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064260/m06426050.png" />-smooth strictly convex body <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064260/m06426051.png" />, the mixed volume <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064260/m06426052.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064260/m06426053.png" />, is equal to the integral of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064260/m06426054.png" />-th elementary symmetric function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064260/m06426055.png" /> of the principal radii of curvature, regarded as a function of the normal to the sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064260/m06426056.png" />. In the case of a general convex body, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064260/m06426057.png" /> is the total value of the measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064260/m06426058.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064260/m06426059.png" />, defined below and called the curvature function. (In the smooth case, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064260/m06426060.png" /> is the density of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064260/m06426061.png" />.) Just as the volume of the body <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064260/m06426062.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064260/m06426063.png" />-th of the integral of its [[Support function|support function]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064260/m06426064.png" /> with respect to its surface function, i.e. the surface area of the image on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064260/m06426065.png" /> under the spherical mapping (cf. [[Spherical map|Spherical map]]), the mixed volume of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064260/m06426066.png" /> bodies can be written as the integral of the support function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064260/m06426067.png" /> of one of them with respect to some measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064260/m06426068.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064260/m06426069.png" /> that depends on the other bodies, called the mixed surface function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064260/m06426070.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/m/m064/m064260/m06426071.png" /></td> </tr></table>

The curvature function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064260/m06426072.png" /> is defined by the equation

<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/m/m064/m064260/m06426073.png" /></td> </tr></table>

(<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064260/m06426074.png" /> occurrences of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064260/m06426075.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064260/m06426076.png" /> occurrences of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064260/m06426077.png" />).

The main content of the theory of mixed volumes is formed by inequalities between mixed volumes (see , [[#References|[3]]]). They include the Minkowski inequality

<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/m/m064/m064260/m06426078.png" /></td> </tr></table>

and the quadratic Minkowski inequality

<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/m/m064/m064260/m06426079.png" /></td> </tr></table>

Both of these are closely connected with the [[Brunn–Minkowski theorem|Brunn–Minkowski theorem]], which is true not only for convex bodies. The Aleksandrov–Fenchel inequality generalizes these, and has the following modification (see ):

<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/m/m064/m064260/m06426080.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/m/m064/m064260/m06426081.png" /></td> </tr></table>

In particular,

<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/m/m064/m064260/m06426082.png" /></td> </tr></table>

A complete system of inequalities characterizing the mixed volume <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064260/m06426083.png" /> has been obtained in the case of two bodies, and more general inequalities have been established (see [[#References|[4]]]).

Many geometric inequalities, such as the classical isoperimetric inequality (cf. [[Isoperimetric inequality, classical|Isoperimetric inequality, classical]]) and several of its refinements, are special cases of inequalities for mixed volumes of convex bodies. The extremum of one of the functionals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064260/m06426084.png" /> when some other such functional is fixed is attained for a sphere. Inequalities in the theory of mixed volumes are used in proving the uniqueness of the solution of the generalized [[Minkowski problem|Minkowski problem]] (see ), in establishing stability in Minkowski problems (see [[#References|[5]]]) and Weil problems (see [[#References|[6]]]), and in the solution of the van der Waerden problem on permanents (see [[#References|[7]]]). Infinite-dimensional analogues of the concepts of the theory of mixed volumes have found application in the theory of Gaussian stochastic processes (see [[#References|[7]]]).

The theory of mixed volumes has deep connections with algebraic geometry. Given a polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064260/m06426085.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064260/m06426086.png" /> complex variables, one can define its Newton polyhedron <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064260/m06426087.png" /> in the following way. To each monomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064260/m06426088.png" /> occurring in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064260/m06426089.png" /> with a non-zero coefficient, one assigns the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064260/m06426090.png" />, and one defines <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064260/m06426091.png" /> as the convex hull of these points. The typical number of solutions of the system of polynomial equations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064260/m06426092.png" /> is equal to the mixed volume of the polyhedron <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064260/m06426093.png" /> divided by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064260/m06426094.png" />. Among other things, this allows one to give an algebraic proof of the Aleksandrov–Fenchel inequality (see [[#References|[10]]]).

In the theory of mixed volumes, a convex body is identified with its support function. This is extended to differences of these functions, and then to arbitrary continuous functions on a sphere (see , [[#References|[9]]]). Using the analogous decomposition of the vector of the centre of gravity of a body <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064260/m06426095.png" /> multiplied by its volume, one can define the so-called mixed direction vectors, which are vector analogues of mixed volumes. The centres of gravity of a body <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064260/m06426096.png" /> coincide up to a multiplicative constant with the mixed direction vectors of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064260/m06426097.png" /> and the sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064260/m06426098.png" /> (see [[#References|[11]]]).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> H. Minkowski, "Theorie der konvexen Körpern, insbesonder der Begründung ihres Oberflächenbegriffs" , ''Gesammelte Abhandlungen'' , '''2''' , Teubner (1911) pp. 131–229</TD></TR><TR><TD valign="top">[2a]</TD> <TD valign="top"> A.D. Aleksandrov, "Zur Theorie gemischter Volumina von konvexen Körpern I. Verallgemeinerungen einiger Begriffe der Theorie von konvexen Körpern" ''Mat. Sb.'' , '''2''' : 5 (1937) pp. 947–972</TD></TR><TR><TD valign="top">[2b]</TD> <TD valign="top"> A.D. Aleksandrov, "Zur Theorie gemischter Volumina von konvexen Körpern II. Neue Ungleichungen zwischen den gemischten Volumina und ihre Anwendungen" ''Mat. Sb.'' , '''2''' : 6 pp. 1205–1238</TD></TR><TR><TD valign="top">[2c]</TD> <TD valign="top"> A.D. Aleksandrov, "Zur Theorie gemischter Volumina von konvexen Körpern III. Die Erweiterung zweier Lehrsätze Minkowski's über die konvexen Polyedern auf die beliebigen konvexen Körper" ''Mat. Sb.'' , '''3''' : 1 (1938) pp. 27–46</TD></TR><TR><TD valign="top">[2d]</TD> <TD valign="top"> A.D. Aleksandrov, "Zur Theorie gemischter Volumina von konvexen Körpern IV. Die gemischten Diskriminanten und die gemischten Volumina" ''Mat. Sb.'' , '''3''' : 2 pp. 227–251</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> H. Busemann, "Convex surfaces" , Interscience (1958) {{MR|0105155}} {{ZBL|0196.55101}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> G. Shephard, "Inequalities between mixed volumes of convex sets" ''Mathematika'' , '''7''' : 14 (1960) pp. 125–138 {{MR|0146736}} {{ZBL|0108.35203}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> V.I. Diskant, "Stability of solutions of Minkowski's equation" ''Sib. Math. J.'' , '''14''' : 3 (1973) pp. 466–469 ''Sibirsk. Mat. Zh.'' , '''14''' : 3 (1973) pp. 669–673 {{MR|333988}} {{ZBL|}} </TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> Yu.A. Volkov, "Estimate of deformation of a convex surface as a function of the change in internal metric" ''Ukrain. Geom. Sb.'' , '''5–6''' (1968) pp. 44–69 (In Russian)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top"> D.E. Knuth, "A permanent inequality" ''Amer. Math. Monthly'' , '''88''' (1981) pp. 731–740 {{MR|0668399}} {{ZBL|0478.15004}} </TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top"> V.N. Sudakov, "Gaussian random processes and measures of solid angles in Hilbert space" ''Soviet Math. Dokl.'' , '''12''' : 2 (1971) pp. 412–415 ''Dokl. Akad. Nauk SSSR'' , '''197''' : 1 (1971) pp. 43–45 {{MR|0288832}} {{ZBL|0231.60025}} </TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top"> H. Busemann, G. Ewald, G. Shepard, "Convex bodies and convexity on Grassmann cones I-IV" ''Math. Ann.'' , '''151''' : 1 (1963) pp. 1–41 {{MR|0157286}} {{MR|0157287}} {{ZBL|0112.37301}} </TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top"> A.G. Khovanskii, "Algebra and mixed volumes" Yu.D. Burago (ed.) V.A. Zalgaller (ed.) , ''Geometric Inequalities'' , Springer (1988) pp. 182–207 (Translated from Russian)</TD></TR><TR><TD valign="top">[11]</TD> <TD valign="top"> R. Schneider, "Krümmungsschwerpunkte konvexer Körper I" ''Abh. Math. Sem. Univ. Hamburg'' , '''37''' (1972) pp. 112–132 {{MR|0307039}} {{ZBL|0229.52005}} </TD></TR></table>



====Comments====


====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> R. Schneider, "On the Aleksandrov–Fenchel inequality" ''Ann. New York Acad. Sci.'' , '''440''' (1985) pp. 132–141 {{MR|0809200}} {{ZBL|0567.52004}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> R. Schneider, "Valuations on convex bodies" P.M. Gruber (ed.) J.M. Wills (ed.) , ''Convexity and its applications'' , Birkhäuser (1983) pp. 170–247 {{MR|0731112}} {{ZBL|0534.52001}} </TD></TR></table>