Journal of the Mathematical Society of Japan

Optimal problem for mixed $p$-capacities

Baocheng ZHU and Xiaokang LUO

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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).

Article information

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

Received: 6 April 2018
First available in Project Euclid: 13 June 2019

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Digital Object Identifier

Primary: 52A20: Convex sets in n dimensions (including convex hypersurfaces) [See also 53A07, 53C45] 53A15: Affine differential geometry

affine surface areas geominimal surface areas isoperimetric inequalities geominimal $p$-capacities


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

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