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 maps and conformal geometry.
A closely related notion is the transfinite diameter or (logarithmic) capacity of a compact simply connected set D, which can be considered as the inverse of the conformal radius of the complement E = Dc viewed from infinity.
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+.
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
Checking that this agrees with the original definition: the uniformizing map f : H → D is and then the derivative can be easily calculated.
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, where dist(z, ∂D) denotes the Euclidean distance between z and the boundary of D, or in other words, the radius of the largest inscribed disk with center z.
Both inequalities are best possible: When D ⊂ C is a connected, simply connected compact set, then its complement E = Dc is a connected, simply connected domain in the Riemann sphere that contains ∞[citation needed], and one can define 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 The coefficient c1 = rad(∞, D) equals the transfinite diameter and the (logarithmic) capacity of D; see Chapter 11 of Pommerenke (1975) and Kuz′mina (2002).
The coefficient c0 is called the conformal center of D. It can be shown to lie in the convex hull of D; moreover, where the radius 2c1 is sharp for the straight line segment of length 4c1.
Let denote the product of pairwise distances of the points
and let us define the following quantity for a compact set D ⊂ C: In other words,
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.
exists and it is called the Fekete constant.
denote the set of all monic polynomials of degree n in C[x], let
with all zeros in D and let us define Then the limits exist and they are called the Chebyshev constant and modified Chebyshev constant, respectively.
Michael Fekete and Gábor Szegő proved that these constants are equal.
The conformal radius is a very useful tool, e.g., when working with the Schramm–Loewner evolution.
A beautiful instance can be found in Lawler, Schramm & Werner (2002).