The Optimal Scaling Parameter in Metric Cotype for Alexandrov Spaces of Nonnegative Curvature

Abstract

We determine the optimal order of the scaling parameter in the metric cotype $2$ inequality for complete Alexandrov spaces of nonnegative curvature. It grows linearly with the dimension $n$ of the discrete torus. This answers Question 17 of Eskenazis, Mendel, and Naor. In their sign-vector normalization on $(\mathbb{Z}/(2m\mathbb{Z}){)}^n$, the square of the optimal metric cotype $2$ constant for this class lies between $\max \{1,n/m\}$ and $e\lceil n/m\rceil$ for every dyadic $m\ge 2$. The upper bound follows from a dyadic Bernoulli-thinning argument using only the Lang--Schroeder--Sturm inequality. Snowflake universality of the fixed Wasserstein space ${\mathcal{P}}_2({\mathbb{R}}^3)$ gives the $n/m$ lower bound.

7 pages