View source for Transfinite diameter

''of a compact set''

A characteristic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093670/t0936701.png" /> of a compact set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093670/t0936702.png" /> in the complex plane serving as a geometric interpretation of the [[Capacity|capacity]] of this set. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093670/t0936703.png" /> be a compact infinite set in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093670/t0936704.png" />-plane. Then 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/t/t093/t093670/t0936705.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</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/t/t093/t093670/t0936706.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093670/t0936707.png" /> is the Euclidean distance between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093670/t0936708.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093670/t0936709.png" />, is called the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093670/t09367011.png" />-th diameter of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093670/t09367012.png" />. In particular, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093670/t09367013.png" /> is the Euclidean diameter of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093670/t09367014.png" />. The points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093670/t09367015.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093670/t09367016.png" /> for which the maximum on the right-hand side of (1) is realized are called the Fekete points (or Vandermonde nodes) for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093670/t09367017.png" />. The sequence of quantities <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093670/t09367018.png" /> is non-increasing: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093670/t09367019.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093670/t09367020.png" /> so that the following limit exists:

<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/t/t093/t093670/t09367021.png" /></td> </tr></table>

The quantity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093670/t09367022.png" /> is also called the transfinite diameter of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093670/t09367023.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093670/t09367024.png" /> is a finite set, then one has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093670/t09367025.png" />. The transfinite diameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093670/t09367026.png" />, the [[Chebyshev constant|Chebyshev constant]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093670/t09367027.png" /> and the capacity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093670/t09367028.png" /> are equal:

<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/t/t093/t093670/t09367029.png" /></td> </tr></table>

The transfinite diameter of a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093670/t09367030.png" /> has the following properties: 1) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093670/t09367031.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093670/t09367032.png" />; 2) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093670/t09367033.png" /> is a fixed complex number and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093670/t09367034.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093670/t09367035.png" />; 3) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093670/t09367036.png" /> is the set of points at a distance at most <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093670/t09367037.png" /> from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093670/t09367038.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093670/t09367039.png" />; 4) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093670/t09367040.png" /> is the set of roots of 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/t/t093/t093670/t09367041.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093670/t09367042.png" /> is a given polynomial and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093670/t09367043.png" /> runs through <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093670/t09367044.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093670/t09367045.png" />. The transfinite diameter of a circle is equal to its radius; the transfinite diameter of a line segment is equal to a quarter of its length.

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093670/t09367046.png" /> be a bounded continuum and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093670/t09367047.png" /> be the component of the complement of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093670/t09367048.png" /> with respect to the extended plane that contains the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093670/t09367049.png" />. Then the transfinite diameter of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093670/t09367050.png" /> is equal to the conformal radius of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093670/t09367051.png" /> (with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093670/t09367052.png" />; cf. [[Conformal radius of a domain|Conformal radius of a domain]]).

The corresponding notions for sets in the hyperbolic and elliptic planes are defined as follows. Consider as a model of the hyperbolic plane the disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093670/t09367053.png" /> with metric defined by the line element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093670/t09367054.png" /> and suppose that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093670/t09367055.png" /> is a closed infinite set in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093670/t09367056.png" />. Then the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093670/t09367058.png" />-th hyperbolic diameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093670/t09367059.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093670/t09367060.png" /> is defined by (1) in which

<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/t/t093/t093670/t09367061.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>

is the hyperbolic pseudo-distance between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093670/t09367062.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093670/t09367063.png" />, that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093670/t09367064.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093670/t09367065.png" /> is the hyperbolic distance between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093670/t09367066.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093670/t09367067.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093670/t09367068.png" /> (see [[Hyperbolic metric|Hyperbolic metric]]). As in the Euclidean case, the sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093670/t09367069.png" /> is non-increasing and the following limit exists:

<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/t/t093/t093670/t09367070.png" /></td> </tr></table>

It is called the hyperbolic transfinite diameter of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093670/t09367071.png" />. Define the hyperbolic Chebyshev constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093670/t09367072.png" /> and the hyperbolic capacity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093670/t09367073.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093670/t09367074.png" /> via the hyperbolic pseudo-distance (2) between the points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093670/t09367075.png" /> by analogy with the Chebyshev constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093670/t09367076.png" /> and capacity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093670/t09367077.png" /> defined via the Euclidean distance between points of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093670/t09367078.png" />-plane. Then one obtains

<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/t/t093/t093670/t09367079.png" /></td> </tr></table>

The hyperbolic transfinite diameter is invariant under the full group of hyperbolic isometries. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093670/t09367080.png" /> is a continuum, then there is a simple relationship between the hyperbolic transfinite diameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093670/t09367081.png" /> and conformal mapping. Namely, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093670/t09367082.png" /> be a continuum in the disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093670/t09367083.png" /> such that the complement of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093670/t09367084.png" /> with respect to this disc is conformally equivalent to the annulus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093670/t09367085.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093670/t09367086.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093670/t09367087.png" />.

Consider as a model of the elliptic plane the extended complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093670/t09367088.png" />-plane with the metric of its Riemann sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093670/t09367089.png" /> of diameter 1, tangent to the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093670/t09367090.png" />-plane at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093670/t09367091.png" />, that is, the metric defined by the line element

<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/t/t093/t093670/t09367092.png" /></td> </tr></table>

furthermore, let the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093670/t09367093.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093670/t09367094.png" /> be identified; these correspond to diametrically-opposite points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093670/t09367095.png" /> under stereographic projection of the extended <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093670/t09367096.png" />-plane onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093670/t09367097.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093670/t09367098.png" /> be a closed infinite set in the extended <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093670/t09367099.png" />-plane, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093670/t093670100.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093670/t093670101.png" />. Then the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093670/t093670103.png" />-th elliptic diameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093670/t093670104.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093670/t093670105.png" /> is defined by (1), in which

<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/t/t093/t093670/t093670106.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>

is the elliptic pseudo-distance between points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093670/t093670107.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093670/t093670108.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093670/t093670109.png" />, that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093670/t093670110.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093670/t093670111.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093670/t093670112.png" />) is the elliptic distance between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093670/t093670113.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093670/t093670114.png" />. As in the previous cases, the sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093670/t093670115.png" /> is non-increasing and the following limit, called the elliptic transfinite diameter of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093670/t093670116.png" />, exists:

<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/t/t093/t093670/t093670117.png" /></td> </tr></table>

Define the elliptic Chebyshev constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093670/t093670118.png" /> and the elliptic capacity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093670/t093670119.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093670/t093670120.png" /> via the elliptic pseudo-distance (3). Then one obtains:

<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/t/t093/t093670/t093670121.png" /></td> </tr></table>

The elliptic transfinite diameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093670/t093670122.png" /> is invariant under the group of fractional-linear transformations

<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/t/t093/t093670/t093670123.png" /></td> </tr></table>

of the extended <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093670/t093670124.png" />-plane onto itself, supplemented by the group of reflections in the elliptic lines. The first of these groups is isomorphic to the group of reflections of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093670/t093670125.png" /> about planes passing through its centre. With this definition the elliptic transfinite diameter of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093670/t093670126.png" /> is related to conformal mapping in the following way. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093670/t093670127.png" /> is a continuum in the extended <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093670/t093670128.png" />-plane, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093670/t093670129.png" />, and the complement of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093670/t093670130.png" /> with respect to the extended plane is conformally equivalent to the annulus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093670/t093670131.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093670/t093670132.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093670/t093670133.png" />.

The notion of the transfinite diameter can be generalized to compacta <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093670/t093670134.png" /> in a multi-dimensional Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093670/t093670135.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093670/t093670136.png" />, and is connected with [[Potential theory|potential theory]]. Let, for points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093670/t093670137.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/t/t093/t093670/t093670138.png" /></td> </tr></table>

be a fundamental solution of the [[Laplace equation|Laplace equation]], and for the set of points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093670/t093670139.png" />, let

<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/t/t093/t093670/t093670140.png" /></td> </tr></table>

Then for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093670/t093670141.png" /> one has

<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/t/t093/t093670/t093670142.png" /></td> </tr></table>

while for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093670/t093670143.png" /> it is expedient (see [[#References|[4]]]) to take

<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/t/t093/t093670/t093670144.png" /></td> </tr></table>

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  M. Fekete,   "Ueber die Verteilung der Wurzeln bei gewissen algebraischen Gleichungen mit ganzzahligen Koeffizienten"  ''Math. Z.'' , '''17'''  (1923)  pp. 228–249</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  G. Pólya,   G. Szegö,   "Ueber den transfiniten Durchmesser (Kapazitätskonstante) von ebenen und räumlichen Punktmengen"  ''J. Reine Angew. Math.'' , '''165'''  (1931)  pp. 4–49</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  G.M. Goluzin,   "Geometric theory of functions of a complex variable" , ''Transl. Math. Monogr.'' , '''26''' , Amer. Math. Soc.  (1969)  (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  L. Carleson,   "Selected problems on exceptional sets" , v. Nostrand  (1967)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  V.I. Smirnov,   A.N. Lebedev,   "Functions of a complex variable" , M.I.T.  (1968)  (Translated from Russian)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  M. Tsuji,   "Potential theory in modern function theory" , Chelsea, reprint  (1959)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  R. Kühnau,   "Geometrie der konformen Abbildung auf der hyperbolischen und der elliptischen Ebene" , Deutsch. Verlag Wissenschaft.  (1974)</TD></TR></table>



====Comments====
Outer radius is another term for transfinite diameter. See [[#References|[a1]]] for a survey on connections between transfinite diameter, [[Robin constant|Robin constant]] and [[Capacity|capacity]] in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093670/t093670145.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093670/t093670146.png" />.

The notion of transfinite diameter also makes good sense in several complex variables, if interpreted in the correct way: (1) with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093670/t093670147.png" /> is a root of a Vandermondian determinant:

<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/t/t093/t093670/t093670148.png" /></td> </tr></table>

where

<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/t/t093/t093670/t093670149.png" /></td> </tr></table>

In <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093670/t093670150.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093670/t093670151.png" /> be an ordered system of monomials of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093670/t093670152.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093670/t093670153.png" /> be a point in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093670/t093670154.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093670/t093670155.png" /> is defined as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093670/t093670156.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093670/t093670157.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093670/t093670158.png" />. The related capacity is the one associated to the complex Monge–Ampère operator.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R.E. Kleinman,   "On a unified characterization of capacity"  J. Král (ed.)  J. Lukeš (ed.)  J. Veselý (ed.) , ''Potential theory. Survey and problems (Prague, 1987)'' , ''Lect. notes in math.'' , '''1344''' , Plenum  (1988)  pp. 103–120</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  M. Klimek,   "Pluripotential theory" , Cambridge Univ. Press  (1991)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  J. Siciak,   "Extremal plurisubharmonic functions and capacities in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093670/t093670159.png" />" , ''Sophia Kokyuroku in Math.'' , '''14''' , Dept. Math. Sophia Univ. Tokyo  (1982)</TD></TR></table>