The dual Orlicz–Brunn–Minkowski theory

Abstract

A first step towards a dual Orlicz–Brunn–Minkowski theory for star sets was taken by Zhu, Zhou, and Xue [44,45]. 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 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.