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Minkowski Problem for Anisotropic p-Torsional Rigidity

A prescribed-measure problem in convex geometry in which the geometric functional is the anisotropic p-torsional rigidity, defined through the solution of the anisotropic (Finsler) p-Laplacian equation on a convex body. One asks: given a Borel measure on the unit sphere, when is it the anisotropic p-torsional measure of some convex body? This paper establishes sufficient and necessary conditions for existence of a solution, and separately solves the log-Minkowski (logarithmic) version of the problem for anisotropic p-torsional rigidity without any symmetry assumption on the measure. The approach follows the variational method of Brunn-Minkowski theory, associating the torsional rigidity with an Lp-type surface area measure and reducing existence to a Monge-Ampere-type equation on the sphere.