The Reverse-log-Brunn-Minkowski inequality

TL;DR

This paper introduces a conjectured reverse-log-Brunn-Minkowski inequality, shows its equivalence to the existing log-Brunn-Minkowski conjecture, and provides new proofs and results for related inequalities in convex geometry.

Contribution

It proposes the RLBM conjecture, establishes the reverse-to-forward principle, and proves the log-Minkowski inequality for zonoids with new equality conditions.

Findings

RLBM conjecture is equivalent to the LBM conjecture.

A simple proof of the LBM inequality in 2D is provided.

The log-Minkowski inequality is proved for zonoids with characterized equality conditions.

Abstract

Firstly, we propose our conjectured Reverse-log-Brunn-Minkowski inequality (RLBM). Secondly, we show that the (RLBM) conjecture is equivalent to the log-Brunn-Minkowski (LBM) conjecture proposed by B\"or\"oczky-Lutwak-Yang-Zhang. We name this as ``reverse-to-forward" principle. Using this principle, we give a very simple new proof of the log-Brunn-Minkowski inequality in dimension two. Finally, we establish the ``reverse-to-forward" principle for the log-Minkowski inequality (LM). Using this principle, we prove the log-Minkowski inequality in the case that one convex body is a zonoid (the inequality part was first proved by van Handle). Via a study of the lemma of relations, the full equality conditions (``dilated direct summands") are also characterized, which turns to be new.