p-capacity vs surface-area

Abstract

This paper is devoted to exploring the relationship between the $[1,n)\ni p$-capacity and the surface-area in ${\mathbb{R}}^{n\ge 2}$ which especially shows: if $\Omega \subset {\mathbb{R}}^n$ is a convex, compact, smooth set with its interior ${\Omega }^{\circ }\ne \varnothing$ and the mean curvature $H(\partial \Omega ,\cdot )>0$ of its boundary $\partial \Omega$ then $${\left(\frac{n(p-1)}{p(n-1)}\right)}^{p-1}\le \frac{\left(\frac{{cap}_p(\Omega )}{(\frac{p-1}{n-p}{)}^{1-p}{\sigma }_{n-1}}\right)}{{\left(\frac{area(\partial \Omega )}{{\sigma }_{n-1}}\right)}^{\frac{n-p}{n-1}}}\le {\left(\sqrt[n-1]{{\int }_{\partial \Omega }(H(\partial \Omega ,\cdot ){)}^{n-1}\frac{d\sigma (\cdot )}{{\sigma }_{n-1}}}\right)}^{p-1}\forall p\in (1,n)$$ whose limits $1\leftarrow p\&p\to n$ imply $$1=\frac{ca{p}_1(\Omega )}{area(\partial \Omega )}\&{\int }_{\partial \Omega }(H(\partial \Omega ,\cdot ){)}^{n-1}\frac{d\sigma (\cdot )}{{\sigma }_{n-1}}\ge 1,$$ thereby not only discovering that the new best known constant is roughly half as far from the one conjectured by Pólya-Szegö in \cite[(2)]{P} but also extending the Pólya-Szegö inequality in \cite[(5)]{P}, with both the conjecture and the inequality being stated for the electrostatic capacity of a convex solid in ${\mathbb{R}}^3$.

13 pages