  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 ${\rm Cap}(K)\geq (n-1)\,H_0{\rm vol}(\partial K)$. 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 ${\rm Cap}(K)\leq
(n-1)\,H_0{\rm vol}(\partial K)$. 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 ${\rm Cap}(K)\geq
(n-1)\,H_0\mathcal{H}^n(\partial K)$ and that equality holds for the round
sphere of radius $H_0^{-1}$.
