The Brunn--Minkowski Inequality and Nontrivial Cycles in the Discrete Torus

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.

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 on the Discrete Hypercube Or: What Is the Ricci Curvature of the Discrete Hypercube? Y. Ollivier and C. Villani

We compare two approaches to Ricci curvature on non-smooth 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 the second author could not be fully implemented. Yet along the way we get new results of a combinatorial and probabilistic natu...

A Brunn-minkowski Inequality for the Integer Lattice

A close discrete analog of the classical Brunn-Minkowksi inequality that holds for finite subsets of the integer lattice is obtained. This is applied to obtain strong new lower bounds for the cardinality of the sum of two finite sets, one of which has full dimension, and, in fact, a method for computing the exact lower bound in this situation, given the dimension of the lattice and the cardinal...