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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 (ℤ/(2mℤ))n, the square of the optimal metric cotype $2$ constant for this class lies between max{1,n/m} and e⌈ n/m⌉ for every dyadic m≥ 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 𝒫2(ℝ3) gives the n/m lower bound.
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BibTeX
@misc{cheng2026optimal,
title = {{The Optimal Scaling Parameter in Metric Cotype for Alexandrov Spaces of Nonnegative Curvature}},
author = {Qingjin Cheng and Longfa Sun and Yipeng Zhang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.08754},
eprint = {2609.08754},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.08754},
}
RIS
TY - UNPB
TI - The Optimal Scaling Parameter in Metric Cotype for Alexandrov Spaces of Nonnegative Curvature
AU - Qingjin Cheng
AU - Longfa Sun
AU - Yipeng Zhang
PY - 2026
DA - 2026/09/09
PB - arXiv
DO - 10.48550/arxiv.2609.08754
UR - https://arxiv.org/abs/2609.08754
AB - 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 (ℤ/(2mℤ))^n, the square of the optimal metric cotype $2$ constant for this class lies between max{1,n/m} and e⌈ n/m⌉ for every dyadic m≥ 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 𝒫_2(ℝ^3) gives the n/m lower bound.
ER -