The $p$-capacitary Orlicz-Hadamard variational formula and Orlicz-Minkowski problems

TL;DR

This paper develops a new geometric theory combining $p$-capacity with Orlicz addition, establishing inequalities and solving the $p$-capacitary Orlicz-Minkowski problem for convex domains.

Contribution

It introduces the $p$-capacitary Orlicz-Brunn-Minkowski theory, defines the Orlicz $L_{}$ mixed $p$-capacity, and solves the associated Minkowski problem under mild conditions.

Findings

Established $p$-capacitary Orlicz-Brunn-Minkowski and Minkowski inequalities.

Introduced the Orlicz $L_{}$ mixed $p$-capacity with geometric interpretation.

Solved the $p$-capacitary Orlicz-Minkowski problem for discrete and general measures.

Abstract

In this paper, combining the $p$-capacity for $p\in (1, n)$ with the Orlicz addition of convex domains, we develop the $p$-capacitary Orlicz-Brunn-Minkowski theory. In particular, the Orlicz $L_{\phi}$ mixed $p$-capacity of two convex domains is introduced and its geometric interpretation is obtained by the $p$-capacitary Orlicz-Hadamard variational formula. The $p$-capacitary Orlicz-Brunn-Minkowski and Orlicz-Minkowski inequalities are established, and the equivalence of these two inequalities are discussed as well. The $p$-capacitary Orlicz-Minkowski problem is proposed and solved under some mild conditions on the involving functions and measures. In particular, we provide the solutions for the normalized $p$-capacitary $L_q$ Minkowski problems with $q>1$ for both discrete and general measures.