## Journal of the Mathematical Society of Japan

### 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.

#### Note

The first author was supported by NSFC (No. 11501185) and the Doctor Starting Foundation of Hubei University for Nationalities (No. MY2014B001).

#### Article information

Source
J. Math. Soc. Japan, Advance publication (2019), 31 pages.

Dates
First available in Project Euclid: 13 June 2019

https://projecteuclid.org/euclid.jmsj/1560391345

Digital Object Identifier
doi:10.2969/jmsj/80268026

#### Citation

ZHU, Baocheng; LUO, Xiaokang. Optimal problem for mixed $p$-capacities. J. Math. Soc. Japan, advance publication, 13 June 2019. doi:10.2969/jmsj/80268026. https://projecteuclid.org/euclid.jmsj/1560391345

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