Abstract
E* Abstract. 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 @K are larger than or equal to H0 > 0, then Cap(K) > (n 1)H0 vol(@K). When K is contained in an (n+1)- dimensional manifold with non-negative Ricci curvature and the mean curvature of @K is smaller than or equal to H0, we prove the inequality Cap(K) 6 (n 1)H0 vol(@K). In both cases we are able to characterize the equality case. Finally, if K is a convex set in Euclidean space R n+1 which admits a supporting sphere of radius H −1 0 at any boundary point, then we prove Cap(K) > (n 1)H0 vol(@K) and that equality holds for the round sphere of radius H −1 0 .
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