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''of a compact set''

A characteristic  $  d = d ( E) $
of a compact set  $  E $
in the complex plane serving as a geometric interpretation of the [[Capacity|capacity]] of this set. Let  $  E $
be a compact infinite set in the  $  z $-
plane. Then the quantity

$$ \tag{1 }
d _ {n} ( E)  = \ 
\left \{
\max _ {z _ {k} , z _ {l} \in E } \ 
\prod _ {1 \leq  k < l \leq  n }
[ z _ {k} , z _ {l} ]
\right \} ^ {2/[ n ( n - 1)] } ,
$$

$$ 
n  =  2, 3 \dots
$$

where  $  [ a, b] = | a - b | $
is the Euclidean distance between  $  a $
and  $  b $, 
is called the  $  n $-
th diameter of  $  E $. 
In particular,  $  d _ {2} ( E) $
is the Euclidean diameter of  $  E $. 
The points  $  z _ {n,1} \dots z _ {n,n} $
of  $  E $
for which the maximum on the right-hand side of (1) is realized are called the Fekete points (or Vandermonde nodes) for  $  E $. 
The sequence of quantities  $  d _ {n} ( E) $
is non-increasing:  $  d _ {n + 1 }  ( E) \leq  d _ {n} ( E) $, 
$  n = 2, 3 \dots $
so that the following limit exists:

$$ 
\lim\limits _ {n \rightarrow \infty }  d _ {n} ( E)  =  d ( E).
$$

The quantity  $  d ( E) $
is also called the transfinite diameter of  $  E $. 
If  $  E $
is a finite set, then one has  $  d ( E) = 0 $. 
The transfinite diameter  $  d ( E) $, 
the [[Chebyshev constant|Chebyshev constant]]  $  \tau ( E) $
and the capacity  $  C ( E) $
are equal:

$$ 
d ( E)  =  \tau ( E)  =  C ( E).
$$

The transfinite diameter of a set  $  E $
has the following properties: 1) if  $  E _ {1} \subset  E $, 
then  $  d ( E _ {1} ) \leq  d ( E) $; 
2) if  $  a $
is a fixed complex number and  $  E _ {1} = \{ {w } : {w = az,  z \in E } \} $, 
then  $  d ( E _ {1} ) = | a | d ( E) $; 
3) if  $  E _  \epsilon  $
is the set of points at a distance at most  $  \epsilon $
from  $  E $, 
then  $  \lim\limits _ {\epsilon \rightarrow 0 }   d ( E _  \epsilon  ) = d ( E) $; 
4) if  $  E  ^ {*} $
is the set of roots of the equation

$$ 
Q ( z)  =  z  ^ {k} + a _ {1} z ^ {k - 1 } + \dots + a _ {k}  =  w,
$$

where  $  Q ( z) $
is a given polynomial and  $  w $
runs through  $  E $, 
then  $  d ( E  ^ {*} ) = \{ d ( E) \}  ^ {1/k} $. 
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  $  E $
be a bounded continuum and let  $  D $
be the component of the complement of  $  E $
with respect to the extended plane that contains the point  $  \infty $. 
Then the transfinite diameter of  $  E $
is equal to the conformal radius of  $  D $(
with respect to  $  \infty $; 
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  $  | z | < 1 $
with metric defined by the line element  $  ds _ {h} = | dz |/( 1 - | z |  ^ {2} ) $
and suppose that  $  E $
is a closed infinite set in  $  | z | < 1 $. 
Then the  $  n $-
th hyperbolic diameter  $  d _ {n,h} ( E) $
of  $  E $
is defined by (1) in which

$$ \tag{2 }
[ a, b]  = \ 
\left |

\frac{a - b }{1 - \overline{a}\; b }
 \ 
\right |
$$

is the hyperbolic pseudo-distance between  $  a $
and  $  b $, 
that is,  $  [ a, b] =  \mathop{\rm tanh}  \rho _ {h} ( a, b) $, 
where  $  \rho _ {h} ( a, b) $
is the hyperbolic distance between  $  a $
and  $  b $
in  $  | z | < 1 $(
see [[Hyperbolic metric|Hyperbolic metric]]). As in the Euclidean case, the sequence  $  d _ {n,h} ( E) $
is non-increasing and the following limit exists:

$$ 
\lim\limits _ {n \rightarrow \infty }  d _ {n,h} ( E)  =  d _ {h} ( E).
$$

It is called the hyperbolic transfinite diameter of  $  E $. 
Define the hyperbolic Chebyshev constant  $  \tau _ {h} ( E) $
and the hyperbolic capacity  $  C _ {h} ( E) $
of  $  E $
via the hyperbolic pseudo-distance (2) between the points of  $  | z | < 1 $
by analogy with the Chebyshev constant  $  \tau ( E) $
and capacity  $  C ( E) $
defined via the Euclidean distance between points of the  $  z $-
plane. Then one obtains

$$ 
d _ {h} ( E)  =  \tau _ {h} ( E)  =  C _ {h} ( E).
$$

The hyperbolic transfinite diameter is invariant under the full group of hyperbolic isometries. If  $  E $
is a continuum, then there is a simple relationship between the hyperbolic transfinite diameter  $  d _ {n} ( E) $
and conformal mapping. Namely, let  $  E $
be a continuum in the disc  $  | z | < 1 $
such that the complement of  $  E $
with respect to this disc is conformally equivalent to the annulus  $  r < | w | < 1 $, 
$  0 < r < 1 $. 
Then  $  r = d _ {n} ( E) $.

Consider as a model of the elliptic plane the extended complex  $  z $-
plane with the metric of its Riemann sphere  $  K $
of diameter 1, tangent to the  $  z $-
plane at the point  $  z = 0 $, 
that is, the metric defined by the line element

$$ 
ds _ {e}  = \ 

\frac{| dz | }{1 + | z |  ^ {2} }
 ;
$$

furthermore, let the points  $  z $
and  $  z  ^ {*} = - 1/z $
be identified; these correspond to diametrically-opposite points of  $  K $
under stereographic projection of the extended  $  z $-
plane onto  $  K $. 
Let  $  E $
be a closed infinite set in the extended  $  z $-
plane,  $  E \cap E  ^ {*} = \emptyset $, 
where  $  E  ^ {*} = \{ {- 1/z } : {z \in E } \} $. 
Then the  $  n $-
th elliptic diameter  $  d _ {n,e} ( E) $
of  $  E $
is defined by (1), in which

$$ \tag{3 }
[ a, b]  = \ 
\left |

\frac{a - b }{1 + \overline{a}\; b }
 \ 
\right |
$$

is the elliptic pseudo-distance between points  $  a $
and  $  b $
of  $  E $, 
that is,  $  [ a, b] =  \mathop{\rm tan}  \rho _ {e} ( a, b) $, 
where  $  \rho _ {e} ( a, b) $(
$  < \pi /2 $) 
is the elliptic distance between  $  a $
and  $  b $. 
As in the previous cases, the sequence  $  d _ {n,e} ( E) $
is non-increasing and the following limit, called the elliptic transfinite diameter of  $  E $, 
exists:

$$ 
\lim\limits _ {n \rightarrow \infty }  d _ {n,e} ( E)  =  d _ {e} ( E).
$$

Define the elliptic Chebyshev constant  $  \tau _ {e} ( E) $
and the elliptic capacity  $  C _ {e} ( E) $
of  $  E $
via the elliptic pseudo-distance (3). Then one obtains:

$$ 
d _ {e} ( E)  =  \tau _ {e} ( E)  =  C _ {e} ( E).
$$

The elliptic transfinite diameter  $  d _ {e} ( E) $
is invariant under the group of fractional-linear transformations

$$ 
z  \rightarrow \ 

\frac{pz + q }{- \overline{q}\; z + p }
 ,\ \ 
| p |  ^ {2} + | q |  ^ {2} = 1,
$$

of the extended  $  z $-
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  $  K $
about planes passing through its centre. With this definition the elliptic transfinite diameter of  $  E $
is related to conformal mapping in the following way. If  $  E $
is a continuum in the extended  $  z $-
plane,  $  E \cap E  ^ {*} = \emptyset $, 
and the complement of  $  E \cup E  ^ {*} $
with respect to the extended plane is conformally equivalent to the annulus  $  r < | w | < 1/r $, 
$  0 < r < 1 $, 
then  $  r = d _ {e} ( E) $.

The notion of the transfinite diameter can be generalized to compacta  $  E $
in a multi-dimensional Euclidean space  $  \mathbf R  ^ {m} $, 
$  m \geq  2 $, 
and is connected with [[Potential theory|potential theory]]. Let, for points  $  x \in \mathbf R  ^ {m} $,

$$ 
H (| x |)  = \ 
\left \{

\begin{array}{ll}
 \mathop{\rm ln}  {
\frac{1}{| x | }
 }  & \textrm{ for }  m = 2,  \\
{
\frac{1}{| x | ^ {m - 2 } }
 }  & \textrm{ for }  m \geq  3,  \\
\end{array}

 \right .$$

be a fundamental solution of the [[Laplace equation|Laplace equation]], and for the set of points  $  ( x _ {j} ) _ {j = 1 }   ^ {n} \subset  E $, 
let

$$ 
\chi _ {n} ( E)  = \ 
\inf
\left \{ { {
\frac{2}{n ( n - 1) }
 }
\sum _ {\begin{array}{c}
j, k = 1 \\
 j < k 
\end{array}
 } ^ { n } 
H (| x _ {j} - x _ {k} |) } : {
( x _ {j} ) _ {j = 1 }   ^ {n} \subset  E
 } \right \}
.
$$

Then for  $  m = 2 $
one has

$$ 
d ( E)  =  C ( E)  = \ 
 \mathop{\rm exp} \left ( - \lim\limits _ {n \rightarrow \infty }  \chi _ {n} ( E) \right ) ;
$$

while for  $  m \geq  3 $
it is expedient (see [[#References|[4]]]) to take

$$ 
d ( E)  =  C ( E)  = \ 
{
\frac{1}{\lim\limits _ {n \rightarrow \infty }  \chi _ {n} ( E) }
 } .
$$

====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  $  \mathbf R  ^ {2} $
or  $  \mathbf R  ^ {n} $.

The notion of transfinite diameter also makes good sense in several complex variables, if interpreted in the correct way: (1) with  $  [ a, b] = | a- b | $
is a root of a Vandermondian determinant:

$$ 
d _ {n} ( E)  =  ( \max _ {x ^ {( n ) }
\in E  ^ {n} } | V ( x  ^ {(} n) ) | ) ^ {2/n( n- 1) } ,
$$

where

$$ 
V( x  ^ {(} n) )  =   \mathop{\rm det}
[ x _ {i}  ^ {j} ] _ {\begin{array} {c}
i = 1 \dots n \\
 j= 0 \dots n- 1 
\end{array}
 } .
$$

In  $  \mathbf C  ^ {n} $, 
let  $  e _ {1} \dots e _ {m _ {n}  } $
be an ordered system of monomials of degree  $  \leq  n $
and let  $  x  ^ {(} n) $
be a point in  $  E ^ {m _ {n} } \subset  \mathbf C ^ {m _ {n} } $. 
Then  $  V ( x  ^ {(} n) ) $
is defined as  $   \mathop{\rm det} [ e _ {i} ( x _ {j} )] $, 
$  x  ^ {n} = ( x _ {1} \dots x _ {m _ {n}  } ) $, 
and  $  d _ {n} ( E) = ( \max _ {x  ^ {(}  n) \in E ^ {m _ {n} } } V( x  ^ {(} n) ) ) ^ {1/  \mathop{\rm deg}  V( x  ^ {n} ) } $. 
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>