Orlicz hormonic Blaschke addition

TL;DR

This paper extends the Orlicz-Brunn-Minkowski theory to dual settings, introducing Orlicz hormonic Blaschke addition and establishing related inequalities, connecting to longstanding conjectures in convex geometry.

Contribution

It introduces the Orlicz hormonic Blaschke addition and derives new dual Minkowski and Brunn-Minkowski inequalities, expanding the theoretical framework of convex geometric analysis.

Findings

Established Orlicz dual Brunn-Minkowski inequalities.

Derived dual Minkowski inequalities related to the log-Brunn-Minkowski conjecture.

Introduced the concept of Orlicz dual projection body.

Abstract

Recently, Gardner, Hug and Weil have introduced the Orlicz-Brunn-Minkowski theory: a general framework, additions, and inequalities. Following this, in the paper we consider Orlicz dual Brunn-Minkowski theory. We introduce Orlicz hormonic Blaschke addition which is an extension of the Lp hormonic Blaschke addition and L p radial Minkowski addition, respectively. Inequalities of dual Minkowski and Brunn-Minkowski type are obtained for the Orlicz hormonic Blaschke addition. The new Orlicz dual Brunn-Minkowski inequality implies the dual and L p -dual Brunn-Minkowski inequalities, respectively. New Orlicz dual Minkowski inequality implies the L p -dual Minkowski inequality. One of these has connections with the conjectured log-Brunn-Minkowski inequality of Lutwak, Yang, and Zhang, and in fact we show a log dual Minkowski inequality. Finally, we introduce the concept of Orlicz dual projection body and an inequality similar to Orlicz projection body is established.