Optimal problem for mixed 𝑝-capacities

Abstract

In this paper, the optimal problem for mixed 𝑝-capacities is investigated. The Orlicz and 𝐿_𝑞 geominimal 𝑝-capacities are proposed and their properties, such as invariance under orthogonal matrices, isoperimetric type inequalities and cyclic type inequalities are provided as well. Moreover, the existence of the 𝑝-capacitary Orlicz–Petty bodies for multiple convex bodies is established, and the Orlicz and 𝐿_𝑞 mixed geominimal 𝑝-capacities for multiple convex bodies are introduced. The continuity of the Orlicz mixed geominimal 𝑝-capacities and some isoperimetric type inequalities of the 𝐿_𝑞 mixed geominimal 𝑝-capacities are proved.