A curved Brunn--Minkowski inequality on the discrete hypercube

TL;DR

This paper explores Ricci curvature on the discrete hypercube, comparing two approaches, and introduces a novel curved Brunn--Minkowski inequality with combinatorial and probabilistic insights.

Contribution

It presents a new curved Brunn--Minkowski inequality on the discrete hypercube, bridging different Ricci curvature approaches and providing novel combinatorial and probabilistic results.

Findings

Positive coarse Ricci curvature on the hypercube

Failure of displacement convexity approach in this setting

New curved Brunn--Minkowski inequality for the hypercube

Abstract

We compare two approaches to Ricci curvature on non-smooth spaces, in the case of the discrete hypercube $\{0,1\}^N$. While the coarse Ricci curvature of the first author readily yields a positive value for curvature, the displacement convexity property of Lott, Sturm and the second author could not be fully implemented. Yet along the way we get new results of a combinatorial and probabilistic nature, including a curved Brunn--Minkowski inequality on the discrete hypercube.