The Lp 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 $(1<p<2)$.