A Curved Brunn–minkowski Inequality on the Discrete Hypercube Or: What Is the Ricci Curvature of the Discrete Hypercube? Y. Ollivier and C. Villani

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}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, includ...

A Curved Brunn Minkowski Inequality for the Symmetric Group

In this paper, we construct an injection A × B → M ×M from the product of any two nonempty subsets of the symmetric group into the square of their midpoint set, where the metric is that corresponding to the conjugacy class of transpositions. If A and B are disjoint, our construction allows to inject two copies of A × B into M ×M . These injections imply a positively curved Brunn-Minkowski inequ...

Ricci curvature of Markov chains on metric spaces

We define the coarse Ricci curvature of metric spaces in terms of how much small balls are closer (in Wasserstein transportation distance) than their centers are. This definition naturally extends to any Markov chain on a metric space. For a Riemannian manifold this gives back, after scaling, the value of Ricci curvature of a tangent vector. Examples of positively curved spaces for this definit...

The Brunn-Minkowski-Type Inequality

Volume difference inequalities for the projection and intersection bodies

In this paper, we introduce a new concept of volumes difference function of the projection and intersection bodies. Following this, we establish the Minkowski and Brunn-Minkowski inequalities for volumes difference function of the projection and intersection bodies.