Proposition 4.2.12AF6AC1F168D7

Proof. By part (3) of proposition 4.1.16, the function $w_ K=-u_ K$ is the Green potential $G_ B\lambda _ K$, and $0\leq w_ K\leq 1$ because $-1\leq u_ K\leq 0$. Since $u_ K$ is the upper semicontinuous regularization of the envelope $f_ K$, which equals $-1$ on $K$, part (2) of theorem 4.1.26 shows that $w_ K=1$ quasi-everywhere on $K$. In particular $I_ B(\lambda _ K)=\int w_ K\, \mathrm d\lambda _ K\leq \lambda _ K(K)$, so $\lambda _ K$ has finite Green energy. As measures of finite Green energy do not charge polar sets, we obtain, with $m=C_ B(K)$,

for every probability measure $\tau $ on $K$ with finite Green energy; in the second identity we used the symmetry of $G_ B$.

Suppose that $m>0$, and let $\tau $ be a probability measure on $K$. If $I_ B(\tau )<\infty $, the Cauchy–Schwarz inequality of proposition 4.2.11 gives $1=I_ B(\lambda _ K,\tau )^2\leq mI_ B(\tau )$; thus $I_ B(\tau )\geq 1/m$ in all cases. Equality holds for $\tau =\lambda _ K/m$, whose energy is $m/m^2=1/m$. Hence the infimum equals $1/m$, which proves the formula. If $\tau _1\neq \tau _2$ were two minimizers, their midpoint would again be a probability measure on $K$, and by positive definiteness

a contradiction. Suppose next that $m=0$. Then $K$ is polar by theorem 4.1.17, so it carries no probability measure of finite energy, and the infimum is $+\infty $ (if $K=\varnothing $, the family is empty).

For the last assertion, recall that balls are regular domains (theorem 3.1.29). The monotonicity of Green functions in the domain (theorem 3.1.34) gives $G_ D\leq G_ B$ on $D\times D$. Hence $I_ B(\tau )\geq I_ D(\tau )$ for every measure $\tau $ supported on a compact set $K\subset D$, and the variational formula, applied in $B$ and in $D$, gives $C_ B(K)\leq C_ D(K)$. Taking suprema over compact subsets, we obtain $C_ B(O)\leq C_ D(O)$ for every open $O\Subset D$. Every such $O$ containing $A$ is also allowed in the infimum defining $C_ B^*(A)$, so $C_ B^*(A)\leq C_ D^*(A)$. $\square$