Logarithmic capacity

Remarks

Example: for the closed disc or the circle of radius $r$, the minimizing measure is normalized arc length on the circle of radius $r$, and $\operatorname{cap} = r$. For a segment of length $L$, $\operatorname{cap} = L/4$. A finite set $K$ has $\operatorname{cap}(K) = 0$: every probability measure on $K$ has an atom, and an atom of mass $m > 0$ at $a$ contributes $m^2 \log(1/\abs{a - a}) = \infty$ to $I(\mu)$. Let $K$ be connected with more than one point, and let its complement in the Riemann sphere $\hat\C$ be simply connected. Let $\varphi$ be the conformal map of $\hat\C \setminus K$ onto $\set{\abs{z} > 1} \cup \set{\infty}$ with $\varphi(\infty) = \infty$. Near $\infty$ it has the form $\varphi(z) = z/c + O(1)$ with $c > 0$, and then $\operatorname{cap}(K) = c$, the outer conformal radius of $K$. By theorems of Fekete and Szegő, the capacity also equals the transfinite diameter of $K$ and its Chebyshev constant.