Positively curved graphs

This paper consists of two halves. In the first half of the paper, we consider real‐valued functions f whose domain is the vertex set of a graph G and that are Lipschitz with respect to the graph distance. By placing a uniform distribution on the vertex set, we treat as a random variable. We investigate the link between the isoperimetric function of G and the functions f that have maximum variance or meet the bound established by the subgaussian inequality. We present several results describing the extremal functions, and use those results to (a) resolve a conjecture by Bobkov, Houdré, and Tetali characterizing the extremal functions of the subgaussian inequality of the odd cycle and (b) provide a construction that gives a partial negative answer to a question by Alon, Boppana, and Spencer on the relationship between maximum variance functions and the isoperimetric function of product graphs. While establishing a discrete analogue of the curved Brunn‐Minkowski inequality for the discrete hypercube, Ollivier and Villani suggested several avenues for research. We resolve them in the second half of the paper as follows: They propose that a bound on t ‐midpoints can be obtained by repeated application of the bound on midpoints, if the original sets are convex. We construct a specific example where this reasoning fails, and then prove our construction is general by characterizing the convex sets in the discrete hypercube. A second proposed technique to bound t ‐midpoints involves new results in concentration of measure. We follow through on this proposal, with heavy use on results from the first half of the paper. We show that the curvature of the discrete hypercube is not positive or zero, but we also give a result indicating that it may satisfy a weaker version of curvature.