The Dual Orlicz-Brunn-Minkowski Theory

TL;DR

This paper develops a new dual Orlicz-Brunn-Minkowski theory for star sets, establishing inequalities and formulas that extend classical geometric inequalities and introduce novel addition operations.

Contribution

It introduces the dual Orlicz-Brunn-Minkowski theory, including new addition operations and inequalities, extending classical results in convex geometry.

Findings

Established dual Orlicz-Brunn-Minkowski inequality.

Derived dual Orlicz mixed volume formula.

Connected radial M-addition to Orlicz addition.

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 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.