A Curved Brunn-Minkowski Inequality for the Symmetric Group

TL;DR

This paper establishes a new Brunn-Minkowski inequality for the symmetric group using a novel injection construction, suggesting potential for improved curvature bounds and connecting to concentration inequalities.

Contribution

It introduces a curved Brunn-Minkowski inequality for the symmetric group and proposes a method to improve curvature bounds via concentration inequalities.

Findings

Constructed an injection from A×B to M×M in the symmetric group.

Derived a Brunn-Minkowski inequality with positive curvature for the symmetric group.

Identified a hypothetical concentration inequality that could optimize the curvature bound.

Abstract

In this paper, we construct an injection $A \times B \rightarrow M \times M$ 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 $A \times B$ into $M \times M$. These injections imply a positively curved Brunn-Minkowski inequality for the symmetric group analogous to that obtained by Ollivier and Villani for the hypercube. However, while Ollivier and Villani's inequality is optimal, we believe that the curvature term in our inequality can be improved. We identify a hypothetical concentration inequality in the symmetric group and prove that it yields an optimally curved Brunn-Minkowski inequality.