On the polar Orlicz-Minkowski problems and the $p$-capacitary Orlicz-Petty bodies

Abstract

In this paper, we propose and study the polar Orlicz-Minkowski problems: under what conditions on a nonzero finite measure $\mu$ and a continuous function $\phi :(0,\infty )\to (0,\infty )$, there exists a convex body $K\in {\mathcal{K}}_0$ such that $K$ is an optimizer of the following optimization problems: \begin{equation*} \inf/\sup \bigg\{\int_{S^{n-1}}φ\big( h_L \big) \,d μ: L \in \mathcal{K}_{0} \ \text{and}\ |L^\circ|=ω_{n}\bigg\}. \end{equation*} The solvability of the polar Orlicz-Minkowski problems is discussed under different conditions. In particular, under certain conditions on $\phi ,$ the existence of a solution is proved for a nonzero finite measure $\mu$ on $S^{n-1}$ which is not concentrated on any hemisphere of $S^{n-1}.$ Another part of this paper deals with the $p$-capacitary Orlicz-Petty bodies. In particular, the existence of the $p$-capacitary Orlicz-Petty bodies is established and the continuity of the $p$-capacitary Orlicz-Petty bodies is proved.

This paper has been accepted by Indiana University Mathematics Journal