An inequality \`a la Szeg\H{o}-Weinberger for the $p-$Laplacian on convex sets

In this paper we prove a sharp inequality of Szeg\H{o}-Weinberger type for the first nontrivial eigenvalue of the $p-$Laplacian with Neumann boundary conditions. This applies to convex sets with given diameter. Some variants, extensions and limit cases are investigated as well.