Curvature-Based Bounds for $p$-Capacity and Capacity--Area Inequalities

We establish curvature-based bounds for the $p$-capacity of convex domains and related capacity--area inequalities. Let $K$ denote the Gauss--Kronecker curvature of $\partialΩ$. In Euclidean space, if $ 0<K_-\leq K\leq K_+, $ then, for every $1<p\leq \frac{n+1}{2}$, $$ \operatorname{Cap}_p\left(B(K_