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Difference between revisions of "Transfinite diameter"

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''of a compact set''
 
''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
+
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
 +
 
 +
$$ \tag{1 }
 +
d _ {n} ( E)  = \
 +
\left \{
 +
\max _ {z _ {k} , z _ {l} \in E } \
 +
\prod _ {1 \leq  k < l \leq  n }
 +
[ z _ {k} , z _ {l} ]
 +
\right \} ^ {2/[ n ( n - 1)] } ,
 +
$$
 +
 
 +
$$
 +
= 2, 3 \dots
 +
$$
 +
 
 +
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:
 +
 
 +
$$
 +
\lim\limits _ {n \rightarrow \infty }  d _ {n} ( E)  =  d ( E).
 +
$$
  
<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>
+
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:
  
<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>
+
$$
 +
d ( E)  = \tau ( E)  = C ( E).
 +
$$
  
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:
+
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
  
<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>
+
$$
 +
Q ( z)  = z  ^ {k} + a _ {1} z ^ {k - 1 } + \dots + a _ {k}  = w,
 +
$$
  
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:
+
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.
  
<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>
+
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 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
+
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
  
<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>
+
$$ \tag{2 }
 +
[ a, b]  = \
 +
\left |
  
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.
+
\frac{a - b }{1 - \overline{a}\; b }
 +
\
 +
\right |
 +
$$
  
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]]).
+
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:
  
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
+
$$
 +
\lim\limits _ {n \rightarrow \infty }  d _ {n,h} ( E)  = d _ {h} ( E).
 +
$$
  
<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>
+
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
  
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:
+
$$
 +
d _ {h} ( E)  = \tau _ {h} ( E)  = C _ {h} ( E).
 +
$$
  
<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>
+
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) $.
  
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
+
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
  
<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>
+
$$
 +
ds _ {e}  = \
  
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" />.
+
\frac{| dz | }{1 + | z |  ^ {2} }
 +
;
 +
$$
  
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
+
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
  
<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>
+
$$ \tag{3 }
 +
[ a, b]  = \
 +
\left |
  
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
+
\frac{a - b }{1 + \overline{a}\; b }
 +
\
 +
\right |
 +
$$
  
<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  $  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:
  
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:
+
$$
 +
\lim\limits _ {n \rightarrow \infty }  d _ {n,e} ( E)  = d _ {e} ( E).
 +
$$
  
<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  $  \tau _ {e} ( E) $
 +
and the elliptic capacity  $  C _ {e} ( E) $
 +
of  $  E $
 +
via the elliptic pseudo-distance (3). Then one obtains:
  
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:
+
$$
 +
d _ {e} ( E)  = \tau _ {e} ( E)  = C _ {e} ( E).
 +
$$
  
<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  $  d _ {e} ( E) $
 +
is invariant under the group of fractional-linear transformations
  
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
+
$$
 +
z  \rightarrow \
  
<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>
+
\frac{pz + q }{- \overline{q}\; z + p }
 +
,\ \
 +
| p |  ^ {2} + | q |  ^ {2} = 1,
 +
$$
  
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" />.
+
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 <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" />,
+
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} $,
  
<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>
+
$$
 +
H (| x |)  = \
 +
\left \{
  
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
+
be a fundamental solution of the [[Laplace equation|Laplace equation]], and for the set of points $  ( x _ {j} ) _ {j = 1 }  ^ {n} \subset  E $,  
 +
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>
+
$$
 +
\chi _ {n} ( E)  = \
 +
\inf
 +
\left \{ { {
 +
\frac{2}{n ( n - 1) }
 +
}
 +
\sum _ {\begin{array}{c}
 +
j, k = 1 \\
 +
j < k
 +
\end{array}
 +
} ^ { n }
 +
H (| x _ {j} - x _ {k} |) } : {
 +
( x _ {j} ) _ {j = 1 }  ^ {n} \subset  E
 +
} \right \}
 +
.
 +
$$
  
Then for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093670/t093670141.png" /> one has
+
Then for $  m = 2 $
 +
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>
+
$$
 +
d ( E)  = C ( E)  = \
 +
\mathop{\rm exp} \left ( - \lim\limits _ {n \rightarrow \infty }  \chi _ {n} ( E) \right ) ;
 +
$$
  
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
+
while for $  m \geq  3 $
 +
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>
+
$$
 +
d ( E)  = C ( E)  = \
 +
{
 +
\frac{1}{\lim\limits _ {n \rightarrow \infty }  \chi _ {n} ( E) }
 +
} .
 +
$$
  
 
====References====
 
====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>
 
<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====
 
====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" />.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093670/t093670147.png" /> is a root of a Vandermondian determinant:
+
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:
  
<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>
+
$$
 +
d _ {n} ( E)  = ( \max _ {x ^ {( n ) }
 +
\in E  ^ {n} } | V ( x  ^ {(} n) ) | ) ^ {2/n( n- 1) } ,
 +
$$
  
 
where
 
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>
+
$$
 +
V( x  ^ {(} n) )  =   \mathop{\rm det}
 +
[ x _ {i}  ^ {j} ] _ {\begin{array} {c}
 +
i = 1 \dots n \\
 +
j= 0 \dots n- 1
 +
\end{array}
 +
} .
 +
$$
  
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.
+
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====
 
====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>
 
<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>

Revision as of 08:26, 6 June 2020


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 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} $,

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

} .

$$ ===='"`UNIQ--h-0--QINU`"'References==== <table><tr><td valign="top">[1]</td> <td valign="top"> M. Fekete, "Ueber die Verteilung der Wurzeln bei gewissen algebraischen Gleichungen mit ganzzahligen Koeffizienten" ''Math. Z.'' , '''17''' (1923) pp. 228–249</td></tr><tr><td valign="top">[2]</td> <td valign="top"> G. Pólya, G. Szegö, "Ueber den transfiniten Durchmesser (Kapazitätskonstante) von ebenen und räumlichen Punktmengen" ''J. Reine Angew. Math.'' , '''165''' (1931) pp. 4–49</td></tr><tr><td valign="top">[3]</td> <td valign="top"> G.M. Goluzin, "Geometric theory of functions of a complex variable" , ''Transl. Math. Monogr.'' , '''26''' , Amer. Math. Soc. (1969) (Translated from Russian)</td></tr><tr><td valign="top">[4]</td> <td valign="top"> L. Carleson, "Selected problems on exceptional sets" , v. Nostrand (1967)</td></tr><tr><td valign="top">[5]</td> <td valign="top"> V.I. Smirnov, A.N. Lebedev, "Functions of a complex variable" , M.I.T. (1968) (Translated from Russian)</td></tr><tr><td valign="top">[6]</td> <td valign="top"> M. Tsuji, "Potential theory in modern function theory" , Chelsea, reprint (1959)</td></tr><tr><td valign="top">[7]</td> <td valign="top"> R. Kühnau, "Geometrie der konformen Abbildung auf der hyperbolischen und der elliptischen Ebene" , Deutsch. Verlag Wissenschaft. (1974)</td></tr></table> ===='"`UNIQ--h-1--QINU`"'Comments==== Outer radius is another term for transfinite diameter. See [[#References|[a1]]] for a survey on connections between transfinite diameter, [[Robin constant|Robin constant]] and [[Capacity|capacity]] in $ \mathbf R ^ {2} $ or $ \mathbf R ^ {n} $. The notion of transfinite diameter also makes good sense in several complex variables, if interpreted in the correct way: (1) with $ [ a, b] = | a- b | $ is a root of a Vandermondian determinant: $$ 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

[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)
How to Cite This Entry:
Transfinite diameter. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Transfinite_diameter&oldid=49009
This article was adapted from an original article by G.V. Kuz'minaE.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article