Robin constant and logarithmic capacity of a compact set

Let $K\subseteq \mathbb{C}$ be compact and nonempty, and let $P(K)$ be the set of Borel probability measures on $K$ (Probability measures and probability spaces), viewed as measures on $\mathbb{C}$ carried by $K$. Every $\mu \in P(K)$ has compact support contained in $K$, so its logarithmic energy $I(\mu )\in (-\mathrm{\infty },+\mathrm{\infty }]$ is the quantity of Logarithmic potential and energy of a positive compactly supported measure; indeed the kernel $k(z,w)=\log (1\mathrm{/}\mathrm{\mid }z-w\mathrm{\mid })$ is bounded below on $K\times K$: it is $\ge -\log \mathrm{diam}K$ when $\mathrm{diam}K>0$, and $I(\mu )=+\mathrm{\infty }$ for the only measure of $P(K)$ when $K$ is a singleton.

The Robin constant of $K$ is

$V_K:={\inf }_{\mu \in P(K)}I(\mu )\in (-\mathrm{\infty },+\mathrm{\infty }]\mathrm{.}$

The set $P(K)$ is nonempty: it contains the Dirac measure at any point of $K$ (The Dirac set function at a point). Put $E_K:=\{I(\mu ):\mu \in P(K),I(\mu )<+\mathrm{\infty }\}$. If $E_K=\mathrm{\varnothing }$, all energies are $+\mathrm{\infty }$ and we set $V_K=+\mathrm{\infty }$. Otherwise $K$ is not a singleton, and $E_K$ is a nonempty subset of $\mathbb{R}$ bounded below by $-\log \mathrm{diam}K$, so Every nonempty set bounded below has an infimum gives its real infimum. Set $V_K=\inf E_K$ in this case. Adding the value $+\mathrm{\infty }$ does not change any real lower bound, so this is exactly the extended infimum displayed above. The logarithmic capacity of $K$ is

$\mathrm{cap}(K):=\{\begin{array}{ll}\exp (-V_K)& \text{if }{V}_K<+\mathrm{\infty },\\ 0& \text{if }{V}_K=+\mathrm{\infty },\end{array}$

with $\exp$ the real exponential function (The real exponential function and the number $e$ by a power series). Thus $\mathrm{cap}(K)\in [0,\mathrm{\infty })$ for every nonempty compact $K$, and for the empty set one uses the separate convention

$\mathrm{cap}(\mathrm{\varnothing }):=0.$

Zero capacity as total divergence of the energy. For nonempty compact $K$, $\mathrm{cap}(K)=0$ holds exactly when $V_K=+\mathrm{\infty }$, that is, exactly when $I(\mu )=+\mathrm{\infty }$ for every $\mu \in P(K)$: the set $\{I(\mu ):\mu \in P(K)\}$ lies in $(-\mathrm{\infty },+\mathrm{\infty }]$, so its infimum is $+\mathrm{\infty }$ precisely when all its members are $+\mathrm{\infty }$. Such a set is called polar; this is the defining dichotomy used throughout the page.

Monotonicity under inclusion. If $K\subseteq L$ are nonempty compact sets, then every Borel probability measure on $K$, extended by zero to the Borel subsets of $L$, is a Borel probability measure on $L$ with the same energy; hence $P(K)\subseteq P(L)$, the infimum over the larger set is no larger, $V_L\le V_K$, and $\mathrm{cap}(K)\le \mathrm{cap}(L)$. The explicit case $\mathrm{cap}(\mathrm{\varnothing })=0$ is consistent with this: the empty set is contained in every set.

Normalization. The sign $\exp (-V_K)$ rather than $\exp (V_K)$ is the convention of the cited sources: it makes $\mathrm{cap}(K)$ a length, for instance $\mathrm{cap}\overline{D(a,r)}=r$ for a disc. The constant $V_K$ and the capacity determine each other whenever $V_K<+\mathrm{\infty }$.

Choice. No choice principle is used in this definition. Choosing a point of a nonempty compact set to exhibit nonemptiness of $P(K)$ is a single selection from a nonempty set; the infimum is a set-theoretic construction on a fixed set of extended reals.