  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.
