Best Constants in Poincaré Inequalities for Convex Domains

We prove a Payne-Weinberger type inequality for the p-Laplacian Neumann eigenvalues (p ≥ 2). The inequality provides the sharp upper bound on convex domains, in terms of the diameter alone, of the best constant in Poincaré inequality. The key point is the implementation of a refinement of the classical Pólya-Szegö inequality for the symmetric decreasing rearrangement which yields an optimal weighted Wirtinger inequality.