Transfinite diameter

of a compact set

A characteristic of a compact set in the complex plane serving as a geometric interpretation of the capacity of this set. Let be a compact infinite set in the -plane. Then the quantity

(1)

where is the Euclidean distance between and , is called the -th diameter of . In particular, is the Euclidean diameter of . The points of for which the maximum on the right-hand side of (1) is realized are called the Fekete points (or Vandermonde nodes) for . The sequence of quantities is non-increasing: , so that the following limit exists:

The quantity is also called the transfinite diameter of . If is a finite set, then one has . The transfinite diameter , the Chebyshev constant and the capacity are equal:

The transfinite diameter of a set has the following properties: 1) if , then ; 2) if is a fixed complex number and , then ; 3) if is the set of points at a distance at most from , then ; 4) if is the set of roots of the equation

where is a given polynomial and runs through , then . 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 be a bounded continuum and let be the component of the complement of with respect to the extended plane that contains the point . Then the transfinite diameter of is equal to the conformal radius of (with respect to ; cf. 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 with metric defined by the line element and suppose that is a closed infinite set in . Then the -th hyperbolic diameter of is defined by (1) in which

(2)

is the hyperbolic pseudo-distance between and , that is, , where is the hyperbolic distance between and in (see Hyperbolic metric). As in the Euclidean case, the sequence is non-increasing and the following limit exists:

It is called the hyperbolic transfinite diameter of . Define the hyperbolic Chebyshev constant and the hyperbolic capacity of via the hyperbolic pseudo-distance (2) between the points of by analogy with the Chebyshev constant and capacity defined via the Euclidean distance between points of the -plane. Then one obtains

The hyperbolic transfinite diameter is invariant under the full group of hyperbolic isometries. If is a continuum, then there is a simple relationship between the hyperbolic transfinite diameter and conformal mapping. Namely, let be a continuum in the disc such that the complement of with respect to this disc is conformally equivalent to the annulus , . Then .

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

furthermore, let the points and be identified; these correspond to diametrically-opposite points of under stereographic projection of the extended -plane onto . Let be a closed infinite set in the extended -plane, , where . Then the -th elliptic diameter of is defined by (1), in which

(3)

is the elliptic pseudo-distance between points and of , that is, , where () is the elliptic distance between and . As in the previous cases, the sequence is non-increasing and the following limit, called the elliptic transfinite diameter of , exists:

Define the elliptic Chebyshev constant and the elliptic capacity of via the elliptic pseudo-distance (3). Then one obtains:

The elliptic transfinite diameter is invariant under the group of fractional-linear transformations

of the extended -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 about planes passing through its centre. With this definition the elliptic transfinite diameter of is related to conformal mapping in the following way. If is a continuum in the extended -plane, , and the complement of with respect to the extended plane is conformally equivalent to the annulus , , then .

The notion of the transfinite diameter can be generalized to compacta in a multi-dimensional Euclidean space , , and is connected with potential theory. Let, for points ,

be a fundamental solution of the Laplace equation, and for the set of points , let

Then for one has

while for it is expedient (see [4]) to take

References

[1]	M. Fekete, "Ueber die Verteilung der Wurzeln bei gewissen algebraischen Gleichungen mit ganzzahligen Koeffizienten" Math. Z. , 17 (1923) pp. 228–249
[2]	G. Pólya, G. Szegö, "Ueber den transfiniten Durchmesser (Kapazitätskonstante) von ebenen und räumlichen Punktmengen" J. Reine Angew. Math. , 165 (1931) pp. 4–49
[3]	G.M. Goluzin, "Geometric theory of functions of a complex variable" , Transl. Math. Monogr. , 26 , Amer. Math. Soc. (1969) (Translated from Russian)
[4]	L. Carleson, "Selected problems on exceptional sets" , v. Nostrand (1967)
[5]	V.I. Smirnov, A.N. Lebedev, "Functions of a complex variable" , M.I.T. (1968) (Translated from Russian)
[6]	M. Tsuji, "Potential theory in modern function theory" , Chelsea, reprint (1959)
[7]	R. Kühnau, "Geometrie der konformen Abbildung auf der hyperbolischen und der elliptischen Ebene" , Deutsch. Verlag Wissenschaft. (1974)

Comments

Outer radius is another term for transfinite diameter. See [a1] for a survey on connections between transfinite diameter, Robin constant and capacity in or .

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

where

In , let be an ordered system of monomials of degree and let be a point in . Then is defined as , , and . The related capacity is the one associated to the complex Monge–Ampère operator.

References

[a1]	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
[a2]	M. Klimek, "Pluripotential theory" , Cambridge Univ. Press (1991)
[a3]	J. Siciak, "Extremal plurisubharmonic functions and capacities in " , Sophia Kokyuroku in Math. , 14 , Dept. Math. Sophia Univ. Tokyo (1982)