Comparison results for capacity Ana HurtadoVicente PalmerManuel Ritore 31C1231C1553C2158J6535J25capacityequilibrium potentialhyperbolicitymean curvatureCartan-Hadamard manifolds 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 $\operatorname{Cap}(K) \ge (n-1)H_0\operatorname{vol}(\partial K)$. 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 $\operatorname{Cap}(K) \le (n-1)H_0\operatorname{vol}(\partial K)$. 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 $\operatorname{Cap}(K) \ge (n-1)H_0\mathcal{H}^n(\partial K)$ and that equality holds for the round sphere of radius $H_0^{-1}$. Indiana University Mathematics Journal 2012 text pdf 10.1512/iumj.2012.61.4564 10.1512/iumj.2012.61.4564 en Indiana Univ. Math. J. 61 (2012) 539 - 555 state-of-the-art mathematics http://iumj.org/access/