math.FA — Functional Analysis

Improved scaling in the metric cotype inequality

Contributed by Manor Mendel

Let $2\le q<\infty$. It is proved that every Banach space with Rademacher cotype $q$ satisfies the metric cotype $q$ inequality with scaling parameter $m=O(n^{1/2+1/q})$. The improved bound on the scaling is achieved by reducing the metric cotype problem to a smoothing and approximation criterion on the continuous torus. For a single unconditional kernel, the criterion is controlled by its mean $\ell_\infty$ radius and its variation in the main diagonal direction. The construction fits the universal smoothing and approximation scheme of Giladi, Mendel, and Naor. The proofs were generated by GPT~6 Astra with the help of the author.

Author comments: 17 pages