  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.
