Capacity, convexity, and isoperimetric inequalities
Given a convex body \(K\) in \(\mathbb{R}^ n\), \(n \geq 3\), let \(C(K)\) denote the capacity of \(K\), \(r_ e\) the circumradius of \(K\), \(r_ 0\) the radius of a ball \(K_ 0\) with the same volume as \(K\) and \(d_ e(K):={r_ e \over r_ 0}-1\) the outer spherical deficit of \(K\). Moreover, let \(g(t) = t^ 3/ \log {1 \over t}\) if \(n=3\), \(g(t) = t^{{n + 3 \over 2}}\) if \(n \geq 4\). Recently, the reviewer and \textit{N. Nadirashvili} [Potential Anal. 3, No. 1, 1-14 (1994; reviewed below)] proved that there exist positive constants \(c\) and \(\delta\), depending only on \(n\), such that we have \(C(K)/C(K_ 0)-1 \geq cg(d_ e(K))\) for any convex body \(K\) in \(\mathbb{R}^ n\) for which \(d_ e(K) \leq \delta\). The author turns this isoperimetric inequality into a proper stability result showing that \(d_ e(K) \leq \delta\) whenever \(C(K)/C(K_ 0)-1<cg(\delta)\).
scientific article; zbMATH DE number 495532