Reflexive midpoint convexity and diamond distortion

What OpenAI claims

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 √(1+k/12), so midpoint uniform convexity does not force uniformly bounded diamond distortion even under reflexivity. (Lean)

Papers (7)

1. Asymptotic midpoint uniform convexity and unbounded diamond distortion in a reflexive tree space

   27 September 2026 · Preprint, not peer reviewed · PDF

   We construct a separable reflexive real Banach space whose given norm is asymptotically midpoint uniformly convex but which admits no asymptotically uniformly convex equivalent norm. Its averaged midpoint modulus is at least √(1+t²/12)-1, and the countably branching diamond of depth k has distortion at least √(1+k/12) in this space. This gives a negative answer to the reflexive diamond converse for asymptotic uniform convexifiability.

2. Midpoint lenses in segment spaces

   27 September 2026 · Preprint, not peer reviewed · PDF

   For a real segment-forest dual with unbounded finite component heights and the real infinite-height coordinate predual, we bound the tail of an arbitrary displacement in a symmetric lens by 2√(R²-‖x‖²), where R is the lens radius and x is its finitely supported center. Both given norms are asymptotically midpoint uniformly convex, although neither space admits an equivalent asymptotically uniformly convex norm. The finite-height forest dual is reflexive.

3. Distortion of countably branching diamonds from midpoint and tree energies

   27 September 2026 · Preprint, not peer reviewed · PDF

   We derive quantitative distortion bounds for countably branching diamond graphs from midpoint estimates and direct tree energies. In the dual of a finite-height segment forest, every distortion-D embedding of the depth-k diamond satisfies D² ≥ 1+k/4. The same bound holds in the infinite-height coordinate predual. We also obtain power-type distortion bounds for path-cost and recursive tree norms.

4. Exact asymptotic moduli in a Daugavet subspace of L1

   27 September 2026 · Preprint, not peer reviewed · PDF

   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.

5. Midpoint convexity from bounded tree potentials and path costs

   27 September 2026 · Preprint, not peer reviewed · PDF

   Four real Banach spaces defined by bounded tree potentials satisfy the averaged asymptotic midpoint bound δ̂(t) ≥ √(1+t²/4)-1 for 0 < t < 1, while none admits an asymptotically uniformly convex (AUC) renorming. On finite-height trees, the globally constrained norm equals the least additive cost of a Hilbert vector and root paths. The quadratic path and segment-start outer Hilbert sums are reflexive and asymptotically midpoint uniformly convex, and admit no equivalent AUC norm.

6. Independent products in real L1: asymptotic midpoint convexity without AUC renormings

   27 September 2026 · Preprint, not peer reviewed · PDF

   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.

7. Midpoint convexity from two recursive potentials

   27 September 2026 · Preprint, not peer reviewed · PDF

   We study tree norms computed by two least nonnegative fields whose difference is the vector. For Euclidean child aggregation, two root-sum spaces have an averaged asymptotic midpoint modulus of at least t³/128 for 0 < t < 1, including a reflexive joining-root space. A reflexive construction with a fixed zero root also satisfies a homogeneous cubic estimate. These spaces admit no equivalent asymptotically uniformly convex norm. We also obtain sixth-power and cubic estimates when the aggregation exponent depends on the height of a finite component.

Status

Preprint. The papers have not been independently reviewed. Treat the result as OpenAI's claim until other mathematicians have checked it. If a video exists, it illustrates the statement; it is not the proof.