The Dual Orlicz-Brunn-Minkowski theory

Richard J. Gardner Western Washington University Daniel Hug Karlsruhe Institute of Technology Wolfgang Weil Karlsruhe Institute of Technology Deping Ye Memorial University of Newfoundland

Convex and Discrete Geometry mathscidoc:1806.40003

J. Math. Anal. Appl., 430, 810-829, 2015
A first step towards a dual Orlicz-Brunn-Minkowski theory for star sets was taken by Zhu, Zhou, and Xue \cite{ZZ, ZZX}. In this essentially independent work we provide a more general framework and results. 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 B\"{o}r\"{o}czky, 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.
compact convex set, star set, star body, Brunn-Minkowski theory, Orlicz-Brunn-Minkowski theory, Minkowski addition, $L_p$ addition, $M$-addition, Orlicz addition, radial addition, Brunn-Minkowski inequality, Minkowski's first inequality
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  • This paper provides the foundation for the promising dual Orlicz-Brunn-Minkowski theory. The authors have been awarded the 2017 JMAA Ames Award.
  title={The Dual Orlicz-Brunn-Minkowski theory},
  author={Richard J. Gardner, Daniel Hug, Wolfgang Weil, and Deping Ye},
  booktitle={J. Math. Anal. Appl.},
Richard J. Gardner, Daniel Hug, Wolfgang Weil, and Deping Ye. The Dual Orlicz-Brunn-Minkowski theory. 2015. Vol. 430. In J. Math. Anal. Appl.. pp.810-829.
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