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Mathematics > Metric Geometry

arXiv:1407.7311 (math)
[Submitted on 28 Jul 2014]

Title:The Dual Orlicz-Brunn-Minkowski Theory

Authors:Richard J. Gardner, Daniel Hug, Wolfgang Weil, Deping Ye
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Abstract:This paper introduces the dual Orlicz-Brunn-Minkowski theory for star sets. A radial Orlicz addition of two or more star sets is proposed and a corresponding dual Orlicz-Brunn-Minkowski inequality is established. Based on a radial Orlicz linear combination of two star sets, a formula for the dual Orlicz mixed volume is derived and a corresponding dual Orlicz-Minkowski inequality proved. The inequalities proved yield as special cases the precise duals of the conjectured log-Brunn-Minkowski and log-Minkowski inequalities of Böröczky, Lutwak, Yang, and Zhang. A new addition of star sets called radial $M$-addition is also introduced and shown to relate to the radial Orlicz addition.
Comments: The present paper is a combination of a manuscript by the first three authors dated May 21, 2013, and the preprint arXiv:1404.6991 written independently by the fourth author. Special cases of some of the results in the present paper were obtained independently in Chapter 5 of the thesis of B. Zhu (see Appendix for a brief description of the difference between them)
Subjects: Metric Geometry (math.MG); Differential Geometry (math.DG)
MSC classes: 52A20, 52A30, 52A39, 52A40
Cite as: arXiv:1407.7311 [math.MG]
  (or arXiv:1407.7311v1 [math.MG] for this version)
  https://doi.org/10.48550/arXiv.1407.7311
Journal reference: J. Math. Anal. Appl., 430 (2015) 810-829
Related DOI: https://doi.org/10.1016/j.jmaa.2015.05.016

Submission history

From: Deping Ye [view email]
[v1] Mon, 28 Jul 2014 02:50:47 UTC (22 KB)
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