p-capacity vs surface-area

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