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| In mathematics, the '''conformal radius''' is a way to measure the size of a [[simply connected]] planar domain ''D'' viewed from a point ''z'' in it. As opposed to notions using [[Euclidean distance]] (say, the radius of the largest inscribed disk with center ''z''), this notion is well-suited to use in [[complex analysis]], in particular in [[conformal map]]s and [[conformal geometry]].
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| A closely related notion is the '''transfinite diameter''' or '''(logarithmic) capacity''' of a [[compact space|compact]] simply connected set ''D'', which can be considered as the inverse of the conformal radius of the [[Complement (set theory)|complement]] ''E'' = ''D<sup>c</sup>'' viewed from [[Riemann sphere|infinity]].
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| ==Definition==
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| Given a simply connected domain ''D'' ⊂ '''C''', and a point ''z'' ∈ ''D'', by the [[Riemann mapping theorem]] there exists a unique conformal map ''f'' : ''D'' → '''D''' onto the [[unit disk]] (usually referred to as the '''uniformizing map''') with ''f''(''z'') = 0 ∈ '''D''' and ''f''′(''z'') ∈ '''R'''<sub>+</sub>. The conformal radius of ''D'' from ''z'' is then defined as
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| : <math>\mathrm{rad}(z,D) := \frac{1}{f'(z)}\,.</math> | |
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| The simplest example is that the conformal radius of the disk of radius ''r'' viewed from its center is also ''r'', shown by the uniformizing map ''x'' ↦ ''x''/''r''. See below for more examples.
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| One reason for the usefulness of this notion is that it behaves well under conformal maps: if φ : ''D'' → ''D''′ is a conformal bijection and ''z'' in ''D'', then <math>\mathrm{rad}(\varphi(z),D') = |\varphi'(z)|\, \mathrm{rad}(z,D)</math>.
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| == A special case: the upper-half plane==
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| Let ''K'' ⊂ '''H''' be a subset of the [[upper half-plane]] such that ''D'' := '''H'''\''K'' is connected and simply connected, and let ''z'' ∈ ''D'' be a point. (This is a usual scenario, say, in the [[Schramm-Loewner evolution]]). By the Riemann mapping theorem, there is a conformal bijection ''g'' : ''D'' → '''H'''. Then, for any such map ''g'', a simple computation gives that
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| : <math>\mathrm{rad}(z,D) = \frac{2\, \mathrm{Im}(g(z))}{|g'(z)|}\,.</math> | |
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| For example, when ''K'' = ∅ and ''z'' = ''i'', then ''g'' can be the identity map, and we get rad(''i'', '''H''') = 2. Checking that this agrees with the original definition: the uniformizing map ''f'' : '''H''' → '''D''' is
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| :<math>f(z)=i\frac{z-i}{z+i},</math> | |
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| and then the derivative can be easily calculated.
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| ==Relation to inradius==
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| That it is a good measure of radius is shown by the following immediate consequence of the [[Schwarz lemma]] and the [[Koebe 1/4 theorem]]: for ''z'' ∈ ''D'' ⊂ '''C''',
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| :<math>\frac{\mathrm{rad}(z,D)}{4} \leq \mathrm{dist} (z,\partial D) \leq \mathrm{rad}(z,D), </math>
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| where dist(''z'', ∂''D'') denotes the Euclidean distance between ''z'' and the [[boundary (topology)|boundary]] of ''D'', or in other words, the radius of the largest inscribed disk with center ''z''.
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| Both inequalities are best possible:
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| : The upper bound is clearly attained by taking ''D'' = '''D''' and ''z'' = 0.
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| : The lower bound is attained by the following “slit domain”: ''D'' = '''C'''\'''R'''<sub>+</sub> and ''z'' = −''r'' ∈ '''R'''<sub>−</sub>. The square root map φ takes ''D'' onto the upper half-plane '''H''', with <math>\varphi(-r) = i\sqrt{r}</math> and derivative <math>|\varphi'(-r)|=\frac{1}{2\sqrt{r}}</math>. The above formula for the upper half-plane gives <math>\mathrm{rad}(i\sqrt{r},\mathbb{H})=2\sqrt{r}</math>, and then the formula for transformation under conformal maps gives rad(−''r'', ''D'') = 4''r'', while, of course, dist(−''r'', ∂''D'') = ''r''.
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| ==Version from infinity: transfinite diameter and logarithmic capacity==
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| When ''D'' ⊂ '''C''' is a simply connected compact set, then its complement ''E'' = ''D<sup>c</sup>'' is a simply connected domain in the [[Riemann sphere]] that contains ∞, and one can define
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| : <math>\mathrm{rad}(\infty,D) := \frac{1}{\mathrm{rad}(\infty,E)} := \lim_{z\to\infty} \frac{f(z)}{z},</math>
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| where ''f'' : '''C'''\'''D''' → ''E'' is the unique bijective conformal map with f(∞) = ∞ and that limit being positive real, i.e., the conformal map of the form
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| :<math>f(z)=c_1z+c_0 + c_{-1}z^{-1} + \dots, \qquad c_1\in\mathbf{R}_+.</math>
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| The coefficient ''c''<sub>1</sub> = rad(∞, ''D'') equals the '''transfinite diameter''' and the '''(logarithmic) capacity''' of ''D''; see Chapter 11 of {{harvtxt|Pommerenke|1975}} and {{harvtxt|Kuz′mina|2002}}. See also the article on the [[capacity of a set]].
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| The coefficient ''c''<sub>0</sub> is called the '''conformal center''' of ''D''. It can be shown to lie in the [[convex hull]] of ''D''; moreover,
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| :<math>D\subseteq \{z: |z-c_0|\leq 2 c_1\}\,,</math>
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| where the radius 2''c''<sub>1</sub> is sharp for the straight line segment of length 4''c''<sub>1</sub>. See pages 12–13 and Chapter 11 of {{harvtxt|Pommerenke|1975}}.
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| ==The Fekete, Chebyshev and modified Chebyshev constants==
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| We define three other quantities that are equal to the transfinite diameter even though they are defined from a very different point of view. Let
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| :<math>d(z_1,\ldots,z_k):=\prod_{1\le i<j\le k} |z_i-z_j|</math>
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| denote the product of pairwise distances of the points <math>z_1,\ldots,z_k</math> and let us define the following quantity for a compact set ''D'' ⊂ '''C''':
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| :<math>d_n(D):=\sup_{z_1,\ldots,z_n\in D} d(z_1,\ldots,z_n)^{\frac{1}{\binom{n}{2}}}</math>
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| In other words, <math>d_n(D)</math> is the supremum of the geometric mean of pairwise distances of ''n'' points in ''D''. Since ''D'' is compact, this supremum is actually attained by a set of points. Any such ''n''-point set is called a '''Fekete set'''.
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| The limit <math>d(D):=\lim_{n\to\infty} d_n(D)</math> exists and it is called the '''Fekete constant'''.
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| Now let <math>\mathcal P_n</math> denote the set of all monic polynomials of degree ''n'' in '''C'''[''x''], let <math>\mathcal Q_n</math> denote the set of polynomials in <math>\mathcal P_n</math> with all zeros in ''D'' and let us define
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| :<math>\mu_n(D):=\inf_{p\in\mathcal P} \sup_{z\in D} |p(z)|</math> and <math>\tilde{\mu}_n(D):=\inf_{p\in\mathcal Q} \sup_{z\in D} |p(z)|</math>
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| Then the limits
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| :<math>\mu(D):=\lim_{n\to\infty} \mu_n(D)^\frac1n</math> and <math>\mu(D):=\lim_{n\to\infty} \tilde{\mu}_n(D)^\frac1n</math>
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| exist and they are called the '''Chebyshev constant''' and '''modified Chebyshev constant''', respectively.
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| [[Michael Fekete]] and [[Gábor Szegő]] proved that these constants are equal.
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| ==Applications==
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| The conformal radius is a very useful tool, e.g., when working with the [[Schramm-Loewner evolution]]. A beautiful instance can be found in {{harvtxt|Lawler|Schramm|Werner|2002}}.
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| ==References==
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| *{{cite book| last=Ahlfors | first=Lars V. | authorlink=Lars Ahlfors | year=1973 | title=Conformal invariants: topics in geometric function theory | series=Series in Higher Mathematics | publisher=McGraw-Hill | isbn= | mr=0357743 | zbl=0272.30012 }}
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| *{{cite book| editor1-last=Horváth | editor1-first=János | year=2005 | title=A Panorama of Hungarian Mathematics in the Twentieth Century, I | series=Bolyai Society Mathematical Studies | publisher=Springer | isbn=3-540-28945-3 }}
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| *{{Citation | last1=Kuz′mina | first1=G. V.|year=2002|title=Conformal radius of a domain | url= http://eom.springer.de/c/c024800.htm}}, from the [[Encyclopaedia of Mathematics]] online.
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| *{{Citation | last1=Lawler | first1=Gregory F. |authorlink1=Gregory Lawler| last2=Schramm | first2=Oded | authorlink2=Oded Schramm| last3=Werner | first3=Wendelin | authorlink3=Wendelin Werner|title= One-arm exponent for critical 2D percolation | url= http://www.math.washington.edu/~ejpecp/viewarticle.php?id=1303&layout=abstract | mr=1887622 | year=2002 | journal= Electronic Journal of Probability | volume=7 | issue=2 | pages=13 pp. | zbl=1015.60091| issn=1083-6489 }}
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| *{{cite book| last=Pommerenke | first= Christian | authorlink=Christian Pommerenke | year=1975 | title=Univalent functions | others=With a chapter on quadratic differentials by Gerd Jensen | publisher=Vandenhoeck & Ruprecht | location=Göttingen | isbn= | zbl=0298.30014 | series=Studia Mathematica/Mathematische Lehrbücher | volume=Band XXV }}
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| ==Further reading==
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| * {{citation | last=Rumely | first=Robert S. | title=Capacity theory on algebraic curves | series=Lecture Notes in Mathematics | volume=1378 | location=Berlin etc. | publisher=[[Springer-Verlag]] | year=1989 | isbn=3-540-51410-4 | zbl=0679.14012 }}
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| ==External links==
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| *{{Citation | last1=Pooh | first1=Charles | title=Conformal radius| url= http://mathworld.wolfram.com/ConformalRadius.html}}. From [[MathWorld]] — A Wolfram Web Resource, created by Eric W. Weisstein.
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| {{DEFAULTSORT:Conformal Radius}}
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| [[Category:Complex analysis]]
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