Transfinite diameter
of a compact set
A characteristic of a compact set E in the complex plane serving as a geometric interpretation of the 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 \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).
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). 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. 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, 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 [4]) to take
d ( E) = C ( E) = \ { \frac{1}{\lim\limits _ {n \rightarrow \infty } \chi _ {n} ( E) } } .
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 \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
[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 ![]() |
Transfinite diameter. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Transfinite_diameter&oldid=49850