A curved Brunn--Minkowski inequality on the discrete hypercube

摘要

We compare two approaches to Ricci curvature on non-smooth spaces, in thecase of the discrete hypercube $\{0,1\}^N$. While the coarse Ricci curvature ofthe first author readily yields a positive value for curvature, thedisplacement convexity property of Lott, Sturm and the second author could notbe fully implemented. Yet along the way we get new results of a combinatorialand probabilistic nature, including a curved Brunn--Minkowski inequality on thediscrete hypercube.