On principal frequencies and isoperimetric ratios in convex sets

On a convex set, we prove that the Poincar'e-Sobolev constant for functions vanishing at the boundary can be bounded from above by the ratio between the perimeter and a suitable power of the $N-$dimensional measure. This generalizes an old result by P'olya. As a consequence, we obtain the sharp {\it Buser’s inequality} (or reverse Cheeger inequality) for the $p-$Laplacian on convex sets. This is valid in every dimension and for every $1<p<+\mathrm{\infty }$. We also highlight the appearing of a subtle phenomenon in shape optimization, as the integrability exponent varies.