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.
The first author was supported by NSFC (No. 11501185) and the Doctor Starting Foundation of Hubei University for Nationalities (No. MY2014B001).
"Optimal problem for mixed $p$-capacities." J. Math. Soc. Japan 71 (4) 1049 - 1079, October, 2019. https://doi.org/10.2969/jmsj/80268026