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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 , so midpoint uniform convexity does not force uniformly bounded diamond distortion even under reflexivity.
Manuscripts in this family · 7
Asymptotic midpoint uniform convexity and unbounded diamond distortion in a reflexive tree space
Sep 27, 2026·manuscript.pdfLean
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 , and the countably branching diamond of depth k has distortion at least in this space. This gives a negative answer to the reflexive diamond converse for asymptotic uniform convexifiability.
Midpoint lenses in segment spaces
Sep 27, 2026·manuscript.pdfLean
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 , 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.
Distortion of countably branching diamonds from midpoint and tree energies
Sep 27, 2026·manuscript.pdfLean
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 . The same bound holds in the infinite-height coordinate predual. We also obtain power-type distortion bounds for path-cost and recursive tree norms.
Exact asymptotic moduli in a Daugavet subspace of L1
Sep 27, 2026·manuscript.pdfLean
For an infinite-dimensional real subspace of L 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 and , 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.
Midpoint convexity from bounded tree potentials and path costs
Sep 27, 2026·manuscript.pdfLean
Four real Banach spaces defined by bounded tree potentials satisfy the averaged asymptotic midpoint bound for , 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.
Independent products in real L1: asymptotic midpoint convexity without AUC renormings
Sep 27, 2026·manuscript.pdfLean
For a countably branching tree, the closed real L 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.
Midpoint convexity from two recursive potentials
Sep 27, 2026·manuscript.pdfLean
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 for , 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.