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We derive sharp upper bounds for eigenvalues of the Laplacian under Neumann boundary conditions on convex domains with given diameter in Euclidean space. We use the Brunn-Minkowski theorem in order to reduce the problem to a question about eigenvalues of certain classes of Sturm-Liouville problems.doi:10.1090/s0002-9939-99-04804-2 fatcat:mlwxej6fgjaerd3csie5irjvrq