Let A be the class of complete Alexandrov spaces of nonnegative curvature. For integers n≥ 1 and dyadic m≥ 2, define Gamma_A(n,m) to be the least Gamma such that, for every X∈A and every map f:(Z/(2mZ))^n→ X, there exists varepsilon∈-1,1^n satisfying [ ∑_x∈(Z/(2mZ))^nd_X bigl(f(x+m varepsilon),f(x)bigr)^2≤ Gamma^2m^2∑_x∈(Z/(2mZ))^n∑_j=1^n d_X bigl(f(x+e_j),f(x)bigr)^2. ] Determine the asymptotic order, as a function of n, of the least scaling parameter m=m(n) for which Gamma_A(n,m) is bounded by a universal constant.

Question 17 of Eskenazis, Mendel, and Naor on metric cotype of Alexandrov spaces

Let $\mathcal{A}$ be the class of complete Alexandrov spaces of nonnegative curvature. For integers $n\ge 1$ and dyadic $m\ge 2$ , define $\Gamma_{\mathcal{A}}(n,m)$ to be the least $\Gamma$ such that, for every $X\in\mathcal{A}$ and every map $f:\left(\mathbb{Z}/(2m\mathbb{Z})\right)^n\to X$ , there exists $\varepsilon\in\{-1,1\}^n$ satisfying

\sum_{x\in(\mathbb{Z}/(2m\mathbb{Z}))^n}d_X\bigl(f(x+m\varepsilon),f(x)\bigr)^2\le \Gamma^2m^2\sum_{x\in(\mathbb{Z}/(2m\mathbb{Z}))^n}\sum_{j=1}^n d_X\bigl(f(x+e_j),f(x)\bigr)^2.

Determine the asymptotic order, as a function of $n$ , of the least scaling parameter $m=m(n)$ for which $\Gamma_{\mathcal{A}}(n,m)$ is bounded by a universal constant.

References

Primary source

arXiv

Additional references

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

banach-space-theory metric-geometry

Posted 4 weeks ago

Progress summary

Refreshed 4 weeks ago

Claimed solved

A September 2026 unrefereed preprint claims to settle the optimal scale in this question, but the result has not been independently verified.

Question 17 asks for the order of growth of the optimal scaling parameter governing metric cotype for nonnegatively curved Alexandrov spaces. The latest report says matching upper and lower bounds have been obtained.

September 2026 claimed resolution

A preprint titled The Optimal Scaling Parameter in Metric Cotype for Alexandrov Spaces of Nonnegative Curvature claims matching bounds in the normalization of Question 17, which would resolve its order-of-growth aspect. The claim is unrefereed and remains unverified.

Current status (as of September 2026): The order-of-growth question is claimed solved by an unrefereed preprint, but independent verification is not documented.

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