Sharp moment-dependent metric cotype scaling for arbitrary Banach spaces

Determine whether every Banach space of Rademacher cotype q satisfies the sharp scaling bound m_q^{(p)}(X;n,Γ)≤Kn^{1/q} for all 1≤p≤q, with constants Γ and K independent of n.

Background

Metric cotype inequalities use a discrete torus and a scale parameter m to characterize Banach-space cotype through distances. The quantity m_q{(p)}(X;n,\Gamma) is the smallest even torus side length for which the moment-exponent-p version of the metric cotype inequality holds in dimension n. The optimal general lower bound has order n{1/q}.

The paper states that Mendel and Naor established sharp metric cotype for K-convex Banach spaces, whereas the general known upper bound was substantially larger. The authors resolve the p=q=2 case for the Schatten trace class S_1, but do not resolve the stated problem for arbitrary Banach spaces, arbitrary admissible p, and arbitrary cotype q.

References

The following open question raised in Section~8, item~1 is one of the fundamental problems in metric cotype theory of Banach spaces. For arbitrary Banach space $X$ of cotype $q$ and every $1\leq p\leq q$, whether there are constants $\Gamma,K<\infty$, independent of $n$, such that $m_q{(p)}(X;n,\Gamma)\leq Kn{1/q}$ for every $n$.

— The Sharp Scale Inequality for the Schatten Trace Class  (2610.21445 - Jiao et al., 15 Oct 2026) in Section 1, Introduction, immediately after the definition of m_q^{(p)}(\mathcal M;n,\Gamma)