The Minkowski problem for the torsional rigidity

TL;DR

This paper establishes existence and uniqueness results for a Minkowski problem related to torsional rigidity in convex geometry, using variational methods and inequalities.

Contribution

It introduces a Minkowski problem for torsional rigidity and proves existence and uniqueness of solutions in convex sets.

Findings

Existence of solutions via variational methods.

Uniqueness follows from Brunn--Minkowski inequality.

Solutions are unique up to translations.

Abstract

We prove the existence and uniqueness up to translations of the solution to a Minkowski type problem for the torsional rigidity in the class of open bounded convex subsets of the $n$-dimensional Euclidean space. For the existence part we apply the variational method introduced by Jerison for analogous problems concerning other variational functionals. Uniqueness follows from the Brunn--Minkowski inequality for the torsional rigidity and corresponding equality conditions.