A Curved Brunn--Minkowski Inequality on the Discrete Hypercube, Or: What Is the Ricci Curvature of the Discrete Hypercube?

Abstract

We compare two approaches to Ricci curvature on nonsmooth 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 Villani 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.