P-capacity vs surface-area

This paper discovers two relationships between the [1,n)∋p-capacity and the surface-area via the Lebesgue volume and the Willmore energy in Rn respectively, whence showing: if Ω⊂Rn is a convex, compact, smooth set with its interior Ω∘≠∅ and the mean curvature H(∂Ω,⋅)>0 of its boundary ∂Ω then(n(p−1)p(n−1))p−1≤(capp(Ω)(p−1n−p)1−pσn−1)(area(∂Ω)σn−1)n−pn−1≤(∫∂Ω(H(∂Ω,⋅))n−1dσ(⋅)σn−1)p−1n−1 and hencecap1(Ω)area(∂Ω)=1≤∫∂Ω(H(∂Ω,⋅))n−1dσ(⋅)σn−1. Of particular interest is that if p=2<3=n in the last but one inequality thencap2(Ω)≥2−13πarea(∂Ω) where 2−13π is smaller than 222/π (conjectured by Pólya-Szegö in [25, (2)]) while bigger than 22/π (obtained by Pólya-Szegö in [27, p. 165(4)]).