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

On principal frequencies and isoperimetric ratios in convex sets