Sharp bounds for the p-torsion of convex planar domains

Abstract

We obtain some sharp estimates for the $p$-torsion of convex planar domains in terms of their area, perimeter, and inradius. The approach we adopt relies on the use of web functions (i.e. functions depending only on the distance from the boundary), and on the behaviour of the inner parallel sets of convex polygons. As an application of our isoperimetric inequalities, we consider the shape optimization problem which consists in maximizing the $p$-torsion among polygons having a given number of vertices and a given area. A long-standing conjecture by P\'olya-Szeg\"o states that the solution is the regular polygon. We show that such conjecture is true within the subclass of polygons for which a suitable notion of "asymmetry measure" exceeds a critical threshold.