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

In this paper, combining the p -capacity for p ∈ ( 1 , n ) with the Orlicz addition of convex domains, we develop the p -capacitary Orlicz–Brunn–Minkowski theory. In particular, the Orlicz L ϕ 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.