Comparison results for capacity

Abstract

We obtain in this paper bounds for the capacity of a compact set $K$. If $K$ is contained in an $(n+1)$-dimensional Cartan-Hadamard manifold, has smooth boundary, and the principal curvatures of $\partial K$ are larger than or equal to $H_0>0$, then $\mathrm{Cap}(K)\ge (n-1)H_0\mathrm{vol}(\partial K)$. When $K$ is contained in an $(n+1)$-dimensional manifold with non-negative Ricci curvature, has smooth boundary, and the mean curvature of $\partial K$ is smaller than or equal to $H_0$, we prove the inequality $\mathrm{Cap}(K)\le (n-1)H_0\mathrm{vol}(\partial K)$. In both cases we are able to characterize the equality case. Finally, if $K$ is a convex set in Euclidean space ${\mathbb{R}}^{n+1}$ which admits a supporting sphere of radius $H_0^{-1}$ at any boundary point, then we prove $\mathrm{Cap}(K)\ge (n-1)H_0{\mathcal{H}}^n(\partial K)$ and that equality holds for the round sphere of radius $H_0^{-1}$.

Final version. To appear in Indiana Univ. Math. J