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 $\varphi:(0,\infty)\rightarrow(0,\infty)$, there exists a convex body $K\in\mathcal{K}<em>0$ such that $K$ is an optimizer of the following optimization problems: \begin{equation*} \inf/\sup \bigg{\int{S^{n-1}}\varphi\big( h_L \big) \,d \mu: L \in \mathcal{K}{0} \ \text{and}\ |L^\circ|=\omega{n}\bigg}. \end{equation*} The solvability of the polar Orlicz-Minkowski problems is discussed under different conditions. In particular, under certain conditions on $\varphi,$ the existence of a solution is proved for a nonzero finite measure $\mu$ on $S<sup>{n-1}$ which is not concentrated on any hemisphere of $S<sup>{n-1}.$ 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.