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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 ${\\rm Cap}(K)\\leq (n-1)\\,H_0{\\rm vol}(\\partial K)$. 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