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Improved scaling in the metric cotype inequality

Primarily AI-generated textHuman understanding: all partsmath.FA — Functional Analysismath.MG — Metric Geometry

Contributed by Manor Mendel ↗

Version 1 / Nov 10, 2026 / CC BY 4.0

Abstract

Let 2≤q<∞2\le q<\infty. It is proved that every Banach space with Rademacher cotype qq satisfies the metric cotype qq inequality with scaling parameter m=O(n1/2+1/q)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 ℓ∞\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

Provenance statement

The proofs in this paper were obtained by GPT~6 Astra. Most of the writing was done by GPT~6 Astra and GPT~6.1 Sol. The author's contributions consist of posing the questions; guiding the AI toward improved bounds, simpler arguments, and clearer exposition; manually editing the text at some points, and verifying the correctness of the proofs.

Tools used

OpenAI
CodexVersion 6 Astra
OpenAI
CodexVersion 6.1 Sol

References

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  1. Veomett, E.; Wildrick, K. Spaces of small metric cotype. J. Topol. Anal., vol. 2, no. 4, pp. 581–597. 2010DOI

Version history

  1. v1 · 17 pagesSubmitted by Manor Mendel17 pagesCurrentNov 08, 2026