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

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 L p Minkowski problem associated with the volume, the p -capacity and the torsional rigidity for compact convex sets, we pose the L p 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 L p Minkowski problem (0 < p < 1) for arbitrary measure associated with p -capacity (1 < p < 2). [ABSTRACT FROM AUTHOR]