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| We obtain in this paper bounds for the capacity of a compact set $K$ with smooth boundary. If $K$ is contained in an $(n+1)$-dimensional Cartan-Hadamard manifold and the principal curvatures of $\partial K$ are larger than or equal to $H_0>0$, then $Cap(K)\geqslant (n-1)\,H_0\, vol(\partial K)$. When $K$ is contained in an $(n+1)$-dimensional manifold with non-negative Ricci curvature and the mean curvature of $\partial K$ is smaller than or equal to $H_0$, we prove the inequality $Cap(K)\leqslant (n-1)\,H_0\,vol(\partial K)$. 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 $Cap(K)\geqslant (n-1)\,H_0\,vol(\partial K)$ and that equality holds for the round sphere of radius $H_0^{-1}$. |
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