Abstract

Based on the L p -harmonic radial combination, Li and Wang researched the asymmetric L p -harmonic radial bodies, which belong to the asymmetric L p -Brunn-Minkowski theory initiated by Ludwig, Haberl and Schuster. In this paper, combined with Orlicz radial combination, we introduce the asymmetric Orlicz radial bodies and research their properties. Further, we also establish some inequalities for this concept.

Highlights

  • The classical Brunn-Minkowski theory, known as the mixed volume theory, has been thought to be the core of modern geometry. Many significant results such as the Brunn-Minkowski inequality and the Minkowski inequality play a significant role in attacking problems in geometry, random matrices and many other fields

  • We refer the reader to the excellent treatises by Gardner [3] and Schneider [4] for more details

  • The lack of homogeneity in this theory makes the corresponding Orlicz addition of convex bodies hard to present. It was not until 2014 that this obstacle was overcome by Gardner et al in [24], where a general framework is introduced for Orlicz-Brunn-Minkowski theory that contains both the new additions and previously proposed concepts, and makes clear for the first time the relation to Orlicz spaces and norms

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Summary

Introduction

The classical Brunn-Minkowski theory, known as the mixed volume theory, has been thought to be the core of modern (convex) geometry. The lack of homogeneity in this theory makes the corresponding Orlicz addition of convex bodies hard to present (note that previous additions in the L p or classical case are homogeneous) It was not until 2014 that this obstacle was overcome by Gardner et al in [24], where a general framework is introduced for Orlicz-Brunn-Minkowski theory that contains both the new additions and previously proposed concepts, and makes clear for the first time the relation to Orlicz spaces and norms. Inspired by the works of the asymmetric L p -Brunn-Minkowski theory (e.g., [16,17,18,19]), we can do research on the extremum problems and Busemann-Petty problems for asymmetric Orlicz radial bodies, which will enrich and further develop the asymmetric theory. We focus first on the basic properties and extremal inequalities

Preliminaries
Properties of Asymmetric Orlicz Radial Bodies
Proofs of Corollaries
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