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October, 2019 Optimal problem for mixed $p$-capacities
Baocheng ZHU, Xiaokang LUO
J. Math. Soc. Japan 71(4): 1049-1079 (October, 2019). DOI: 10.2969/jmsj/80268026


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.

Funding Statement

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


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Baocheng ZHU. Xiaokang LUO. "Optimal problem for mixed $p$-capacities." J. Math. Soc. Japan 71 (4) 1049 - 1079, October, 2019.


Received: 6 April 2018; Published: October, 2019
First available in Project Euclid: 13 June 2019

zbMATH: 07174395
MathSciNet: MR4023296
Digital Object Identifier: 10.2969/jmsj/80268026

Primary: 52A20 , 53A15

Keywords: affine surface areas , geominimal $p$-capacities , geominimal surface areas , Isoperimetric inequalities

Rights: Copyright © 2019 Mathematical Society of Japan


Vol.71 • No. 4 • October, 2019
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