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Independent products in real L1: asymptotic midpoint convexity without AUC renormings
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Abstract
For a countably branching tree, the closed real L1 spans of products of independent exponential or Gaussian-square multipliers along its paths have positive averaged asymptotic midpoint moduli and admit no equivalent asymptotically uniformly convex norm. The renorming obstruction holds for every positive nonconstant mean-one multiplier with finite second moment.
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Notes
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work in progress
Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.
Formalization
- OpenAI math release, Lean scope notes for family 331: Reflexive midpoint convexity and diamond distortion (lean/docs/331.md, release of 2026-10-06). Describes the release's own Lean formalization; for this paper it covers infinite dimension and the absence of an equivalent AUC norm for general positive nonconstant mean-one finite-second-moment multipliers, as well as the explicit exponential and squared-Gaussian lower bounds for the averaged midpoint modulus. [family-level]
Digests and expositions
- Gradient Damage, Every OpenAI Math Result, Part 4 of 4: L = BPL, Integer Multiplication and 82 More (video compilation, 2026-10-14). Chapter 74 at 25:23 provides a brief visual introduction to the family’s reflexive-space claim. [family-level]
Elsewhere
- Emergent Mind: family 331
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:Independent-products-in-real-L1-asymptotic-midpoint-convexity-without-AUC-renormings-September-27-2026,
author = {{OpenAI}},
title = {{Independent products in real $L^1$: asymptotic midpoint convexity without AUC renormings}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/Independent-products-in-real-L1-asymptotic-midpoint-convexity-without-AUC-renormings-September-27-2026/manuscript.pdf}{OAI:Independent-products-in-real-L1-asymptotic-midpoint-convexity-without-AUC-renormings-September-27-2026}},
year = {2026}
}Lean formalization
OpenAI provides Lean files for this family: lean/docs/331.md. The Lean statements have not been checked by people against the paper's claims.
Family 331: Reflexive midpoint convexity and diamond distortion
Constructs a real reflexive Banach space with an asymptotically midpoint uniformly convex norm but no equivalent asymptotically uniformly convex norm, extending Baudier's separation to reflexive spaces. In the same space, depth-k countably branching diamonds require distortion at least $\sqrt{1+k/12}$, so midpoint uniform convexity does not force uniformly bounded diamond distortion even under reflexivity.
Other papers in this family:
- Asymptotic midpoint uniform convexity and unbounded diamond distortion in a reflexive tree space · September 27, 2026
- Midpoint lenses in segment spaces · September 27, 2026
- Distortion of countably branching diamonds from midpoint and tree energies · September 27, 2026
- Exact asymptotic moduli in a Daugavet subspace of L1 · September 27, 2026
- Midpoint convexity from bounded tree potentials and path costs · September 27, 2026
- Midpoint convexity from two recursive potentials · September 27, 2026