The [formula omitted] Minkowski problem associated with the compatible functional F

Abstract

Motivated by some properties of the geometric measures for compact convex sets in the Brunn–Minkowski theory, such as the properties of the volume, the p-capacity (1<p<n) and the torsional rigidity for compact convex sets, we introduce a more general geometric invariant, called the compatible functional F. Inspired also by the Lp Minkowski problem associated with the volume, the p-capacity and the torsional rigidity for compact convex sets, we pose the Lp Minkowski problem associated with the compatible functional F and prove the existence of the solutions to this problem for p>0. We will show that the volume, the p-capacity (1<p<2) and the torsional rigidity for compact convex sets are the compatible functionals. Thus, as an application, we provide the solution to the Lp Minkowski problem (0<p<1) for arbitrary measure associated with p-capacity (1<p<2).