Mendel–Naor sharp metric cotype question for L1

For $1\le p<\infty$ , even $m\ge 4$ , and $n\in\mathbb N$ , let $A_p(n,m)$ be the least constant such that every function $f:\mathbb Z_m^n\to L_1$ satisfies

\sum_{i=1}^n\mathbb E_{x\in\mathbb Z_m^n}\left\|f\left(x+\frac m2 e_i\right)-f(x)\right\|_1^p\le A_p(n,m)\,\mathbb E_{x\in\mathbb Z_m^n,\,\varepsilon\in\{-1,1\}^n}\left\|f(x+\varepsilon)-f(x)\right\|_1^p.

Determine the optimal order of $A_p(n,m)$ . The conjectured sharp answer is

A_p(n,m)^{1/p}\asymp\left(m^p n^{(1-p/2)_+}+n\right)^{1/p},

with comparison constants independent of $p$ , $m$ , and $n$ . In particular, for $p=2$ the desired order is $A_2(n,m)\asymp m^2+n$ , yielding the sharp metric-cotype scale $m=O(\sqrt n)$ for $L_1$ .

References

Primary source

arXiv

Additional references

Sharp Metric Cotype Inequalities for L1 via Nonlinear Cut Smoothing — arXiv

Posted 4 weeks ago

Progress summary

Refreshed 4 weeks ago

Claimed solved

A September 2026 unrefereed preprint claims to settle the sharp metric-cotype scale for $L_1$ , but the claim has not been independently verified.

The problem asks for the optimal order in the torus inequality governing metric cotype for $L_1$ , including whether the conjectural quadratic scale can be attained.

Known results

Earlier work recorded $L_1$ as unresolved, with the best bound $m\gtrsim n^{3/2}$ ; the conjectured optimum was $m\ge n^{1/q}$ , hence $m\ge\sqrt n$ for cotype $q=2$ .

September 8, 2026 claimed sharp bound

The preprint Sharp Metric Cotype Inequalities for $L_1$ via Nonlinear Cut Smoothing claims the optimal torus-inequality order, including the quadratic case with scaling $m=O(\sqrt n)$ . This would settle the tracked question, but the result is explicitly unrefereed and remains unverified.

Current status (as of September 2026): The conjectural sharp scale $m=O(\sqrt n)$ is claimed in an unrefereed preprint, but independent verification is absent.

Sources

No solutions have been posted yet.