Dual Orlicz Mixed Quermassintegral

Conference paper
First Online:
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 203)

Abstract

We study the dual Orlicz mixed Quermassintegral. For arbitrary monotone continuous function \(\phi \), the dual Orlicz radial sum and dual Orlicz mixed Quermassintegral are introduced. Then the dual Orlicz–Minkowski inequality and dual Orlicz–Brunn–Minkowski inequality for dual Orlicz mixed Quermassintegral are obtained. These inequalities are just the special cases of their \(L_p\) analogues (including cases \(-\infty<p<0\), \(p=0\), \(0<p<1\), \(p=1\), and \(1<p<+\infty \)). These inequalities for \(\phi =\log t\) are related to open problems including log-Minkowski problem and log-Brunn-Minkowski problem. Moreover, the equivalence of the dual Orlicz–Minkowski inequality for dual Orlicz mixed Quermassintegral and dual Orlicz–Brunn–Minkowski inequality for dual Orlicz mixed Quermassintegral is shown.

Keywords

Star body Orlicz radial sum Dual Orlicz mixed Quermassintegral Dual Orlicz–Minkowski inequality Dual Orlicz–Brunn–Minkowski inequality 

2010 Mathematics Subject Classification

Primary: 52A30 52A40 
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Notes

Acknowledgements

This research is supported in part by National Natural Science Foundation of China (Grant No. 11271302) and Fundamental Research Funds for the Central Universities (No. XDJK2016D026). We like to thank referees for helpful suggestions.

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Copyright information

© Springer Nature Singapore Pte Ltd. 2017

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsSouthwest UniversityChongqingChina

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