A sharp upper bound for the first Dirichlet eigenvalue and the growth of the isoperimetric constant of convex domains

We show that as the ratio between the first Dirichlet eigenvalues of a convex domain and of the ball with the same volume becomes large, the same must happen to the corresponding ratio of isoperimetric constants. The proof is based on the generalization to arbitrary dimensions of Polya and Szego's 1951 upper bound for the first eigenvalue of the Dirichlet Laplacian on planar star-shaped domains which depends on the support function of the domain. As a by-product, we also obtain a sharp upper bound for the spectral gap of convex domains.