# A remark on the Alexandrov-Fenchel inequality

## TL;DR

This paper presents a novel complex-geometric proof of the Alexandrov-Fenchel inequality, avoiding toric compactifications by employing Legendre transforms and advanced Hodge-Riemann relations.

## Contribution

It introduces a new proof method using Legendre transforms and mixed Hodge-Riemann relations, expanding the tools available for inequalities in convex and complex geometry.

## Key findings

- Provides a complex-geometric proof of the Alexandrov-Fenchel inequality
- Develops a non-compact version of the Khovanski-Teissier inequality
- Integrates Timorin's mixed Hodge-Riemann bilinear relation into the proof

## Abstract

In this article, we give a complex-geometric proof of the Alexandrov-Fenchel inequality without using toric compactifications. The idea is to use the Legendre transform and develop the Brascamp-Lieb proof of the Pr\'ekopa theorem. New ingredients in our proof include an integration of Timorin's mixed Hodge-Riemann bilinear relation and a mixed norm version of H\"ormander's $L^2$-estimate, which also implies a non-compact version of the Khovanski\u{i}-Teissier inequality.