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

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.