  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 $1<p<+\infty$. We also
highlight the appearing of a subtle phenomenon in shape optimization, as the
integrability exponent varies.
