Semiclassical inequalities for Dirichlet and Neumann Laplacians on convex domains

We are interested in inequalities that bound the Riesz means of the eigenvalues of the Dirichlet and Neumann Laplacians in terms of their semiclassical counterpart. We show that the classical inequalities of Berezin-Li-Yau and Kr\"oger, valid for Riesz exponents $\gamma\geq 1$, extend to certain values $\gamma<1$, provided the underlying domain is convex. We also study the corresponding optimization problems and describe the implications of a possible failure of P\'olya's conjecture for convex sets in terms of Riesz means. These findings allow us to describe the asymptotic behavior of solutions of a spectral shape optimization problem for convex sets.