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

We compare two approaches to Ricci curvature on nonsmooth spaces in the case of the discrete hypercube \0,1. 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.