  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 $\Omega^\circ\not=\emptyset$ and the mean curvature
$H(\partial\Omega,\cdot)>0$ of its boundary $\partial\Omega$ then $$
\left(\frac{n(p-1)}{p(n-1)}\right)^{p-1}\le\frac{\left(\frac{\hbox{cap}_p(\Omega)}{\big(\frac{p-1}{n-p}\big)^{1-p}\sigma_{n-1}}\right)}{\left(\frac{\hbox{area}(\partial\Omega)}{\sigma_{n-1}}\right)^\frac{n-p}{n-1}}\le\left(\sqrt[n-1]{\int_{\partial\Omega}\big(H(\partial\Omega,\cdot)\big)^{n-1}\frac{d\sigma(\cdot)}{\sigma_{n-1}}}\right)^{p-1}\quad\forall\quad
p\in (1,n) $$ whose limits $1\leftarrow p\ \&\ p\rightarrow n$ imply $$
1=\frac{cap_1(\Omega)}{\hbox{area}(\partial\Omega)}\ \ \& \
\int_{\partial\Omega}\big(H(\partial\Omega,\cdot)\big)^{n-1}\frac{d\sigma(\cdot)}{\sigma_{n-1}}\ge
1, $$ thereby not only discovering that the new best known constant is roughly
half as far from the one conjectured by P\'olya-Szeg\"o in \cite[(2)]{P} but
also extending the P\'olya-Szeg\"o inequality in \cite[(5)]{P}, with both the
conjecture and the inequality being stated for the electrostatic capacity of a
convex solid in $\mathbb R^3$.

Differential Geometry