A letter: The log-Brunn-Minkowski inequality for complex bodies

TL;DR

This paper confirms the log-Brunn-Minkowski inequality for complex convex bodies by linking it to complex interpolation and leveraging a theorem by Cordero-Erausquin, providing a significant step in convex geometry.

Contribution

It establishes the validity of the log-Brunn-Minkowski conjecture for complex bodies using complex interpolation techniques and existing theorems.

Findings

The inequality holds for complex convex bodies.

Connection established between the conjecture and complex interpolation.

Utilizes a theorem by Cordero-Erausquin to prove the conjecture.

Abstract

In this short note we explain why the log-Brunn-Minkowski conjecture is correct for complex convex bodies. We do this by relating the conjecture to the notion of complex interpolation, and appealing to a general theorem by Cordero-Erausquin.