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Exact asymptotic moduli in a Daugavet subspace of L1
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Abstract
For an infinite-dimensional real subspace of L1 whose unit ball is totally bounded in measure and whose norm has the Daugavet property, we compute the averaged midpoint and one-sided asymptotic moduli at every unit center. They are $\max\{t/2,t-1\}$ and $\max\{0,t-2\}$, respectively. The same weak-neighborhood geometry excludes every equivalent asymptotically uniformly convex norm. A quantitative realization of the Kadets–Werner construction supplies a space with both hypotheses.
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Notes
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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 the exact formulas $\max(t/2,t-1)$ and $\max(0,t-2)$ at every unit vector, failure of AUC for every equivalent norm, and the construction of a measure-precompact Daugavet subspace on a countable product of intervals. [family-level]
Possibly relevant literature (found by the agent, not checked)
- T. A. Abrahamsen, V. Lima, A. Martiny, Y. Perreau, Asymptotic geometry and Delta-points (arXiv:2203.14528, 2022; Banach J. Math. Anal. 16 (2022)). Studies when asymptotic smoothness or convexity rules out Daugavet- and Δ-points, which may give context for the paper's weak-neighborhood argument excluding equivalent AUC norms.
Elsewhere
- Emergent Mind: family 331
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:Exact-asymptotic-moduli-in-a-Daugavet-subspace-of-L1-September-27-2026,
author = {{OpenAI}},
title = {{Exact asymptotic moduli in a Daugavet subspace of $L^1$}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/Exact-asymptotic-moduli-in-a-Daugavet-subspace-of-L1-September-27-2026/manuscript.pdf}{OAI:Exact-asymptotic-moduli-in-a-Daugavet-subspace-of-L1-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
- Midpoint convexity from bounded tree potentials and path costs · September 27, 2026
- Independent products in real L1: asymptotic midpoint convexity without AUC renormings · September 27, 2026
- Midpoint convexity from two recursive potentials · September 27, 2026