# Conformal radius

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 = D$^c$ viewed from infinity.

## Definition

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+. The conformal radius of D from z is then defined as
    $\operatorname {rad} (z,D):={\frac {1}{f'(z)}}\,.$
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.
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 $\operatorname {rad} (\varphi (z),D')=|\varphi '(z)|\operatorname {rad} (z,D)$.
The conformal radius can also be expressed as $\exp(\xi _{x}(x))$ where $\xi _{x}(y)$ is the harmonic extension of $\log(|x-y|)$ from $\partial D$ to $D$.

## A special case: the upper-half plane

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
    $\operatorname {rad} (z,D)={\frac {2\operatorname {Im} (g(z))}{|g'(z)|}}\,.$
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
    $f(z)=i{\frac {z-i}{z+i}},$
and then the derivative can be easily calculated.

## Relation to inradius

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,
    ${\frac {\operatorname {rad} (z,D)}{4}}\leq \operatorname {dist} (z,\partial D)\leq \operatorname {rad} (z,D),$
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:
    The upper bound is clearly attained by taking D = D and z = 0.
    The lower bound is attained by the following “slit domain”: D = C\R+ and z = −r ∈ R$_−$. The square root map φ takes D onto the upper half-plane H, with $\varphi (-r)=i{\sqrt {r}}$ and derivative $|\varphi '(-r)|={\frac {1}{2{\sqrt {r}}}}$. The above formula for the upper half-plane gives $\operatorname {rad} (i{\sqrt {r}},\mathbb {H} )=2{\sqrt {r}}$, and then the formula for transformation under conformal maps gives rad(−r, D) = 4r, while, of course, dist(−r, ∂D) = r.

## Version from infinity: transfinite diameter and logarithmic capacity

When D ⊂ C is a connected, simply connected compact set, then its complement E = D$^c$ is a connected, simply connected domain in the Riemann sphere that contains ∞, and one can define
    $\operatorname {rad} (\infty ,D):={\frac {1}{\operatorname {rad} (\infty ,E)}}:=\lim _{z\to \infty }{\frac {f(z)}{z}},$
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
    $f(z)=c_{1}z+c_{0}+c_{-1}z^{-1}+\cdots ,\qquad c_{1}\in \mathbf {R} _{+}.$
The coefficient c$_1$ = rad(∞, D) equals the transfinite diameter and the (logarithmic) capacity of D; see Chapter 11 of Pommerenke (1975) and Kuz′mina (2002).
The coefficient c$_0$ is called the conformal center of D. It can be shown to lie in the convex hull of D; moreover,
    $D\subseteq \{z:|z-c_{0}|\leq 2c_{1}\}\,,$
where the radius 2c$_1$ is sharp for the straight line segment of length 4c$_1$. See pages 12–13 and Chapter 11 of Pommerenke (1975).

## The Fekete, Chebyshev and modified Chebyshev constants

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
    $d(z_{1},\ldots ,z_{k}):=\prod _{1\leq i<j\leq k}|z_{i}-z_{j}|$
denote the product of pairwise distances of the points $z_{1},\ldots ,z_{k}$ and let us define the following quantity for a compact set D ⊂ C:
    $d_{n}(D):=\sup _{z_{1},\ldots ,z_{n}\in D}d(z_{1},\ldots ,z_{n})^{1\left/{\binom {n}{2}}\right.}$
In other words, $d_{n}(D)$ 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.
The limit $d(D):=\lim _{n\to \infty }d_{n}(D)$ exists and it is called the Fekete constant.
Now let ${\mathcal {P}}_{n}$ denote the set of all monic polynomials of degree n in C[x], let ${\mathcal {Q}}_{n}$ denote the set of polynomials in ${\mathcal {P}}_{n}$ with all zeros in D and let us define
    $\mu _{n}(D):=\inf _{p\in {\mathcal {P}}_{n}}\sup _{z\in D}|p(z)|$ and ${\tilde {\mu }}_{n}(D):=\inf _{p\in {\mathcal {Q}}_{n}}\sup _{z\in D}|p(z)|$
Then the limits
    $\mu (D):=\lim _{n\to \infty }\mu _{n}(D)^{1/n}$ and $\mu (D):=\lim _{n\to \infty }{\tilde {\mu }}_{n}(D)^{1/n}$
exist and they are called the Chebyshev constant and modified Chebyshev constant, respectively. Michael Fekete and Gábor Szegő proved that these constants are equal.

## Applications

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).