Reflexive midpoint convexity and diamond distortion

1 · The essentials

Two ways of asking a space to be rounded can look similar but lead to different answers. One tests the behavior of midpoints; the other imposes a stronger requirement on the way distances behave. The main manuscript constructs a space that passes the midpoint test but cannot pass the stronger test, even after a compatible change in how lengths are measured. It does this within a more restricted class of spaces, extending a separation attributed in the repository summary to Baudier. The same example also makes increasingly deep, repeatedly branching diamond-shaped networks increasingly difficult to fit without stretching some distances more than others. Thus, passing the midpoint test does not guarantee a common bound on that stretching. The main result has a Lean formalization, and the companion manuscripts develop related examples and quantitative estimates.

A picture to help (an imperfect image, not to be taken literally)

As an imperfect image, imagine fitting a branching diamond-shaped wire network onto a surface while trying to preserve its distances. A local test of how rounded the surface is does not necessarily tell you how much a deeper network must be stretched; here, however, the actual setting is an infinite-dimensional space rather than a physical surface.

Why it matters

The construction separates asymptotic midpoint uniform convexity from the possibility of an asymptotically uniformly convex renorming, even under reflexivity. Its diamond estimate also rules out uniformly bounded distortion for these graphs in the constructed space. Together, these conclusions show that the midpoint condition cannot substitute for the stronger condition in the ways tested by this family.

2 · Going further

The central problem concerns two asymptotic convexity conditions on Banach spaces and whether changing to an equivalent norm can bridge the gap between them. The main manuscript constructs a separable reflexive real Banach space whose given norm is asymptotically midpoint uniformly convex but which has no equivalent asymptotically uniformly convex (AUC) norm. Its averaged midpoint modulus satisfies $δ (t) \geq 1 + t^{2} /12 - 1$ . The repository summary describes this as extending Baudier’s separation to reflexive spaces.

The metric conclusion concerns countably branching diamond graphs. In the main space, a depth- $k$ diamond requires distortion at least $1 + k /12$ , which grows without bound with depth. The main abstract describes this as a negative answer to the reflexive diamond converse for asymptotic uniform convexifiability. At the level explained by the abstracts, the quantitative approach uses midpoint estimates and tree energies. The lens manuscript bounds the tail of a displacement in a symmetric lens by $2 R^{2} - ∥ x ∥^{2}$ , with lens radius $R$ and finitely supported center $x$ . The diamond manuscript obtains the stronger estimate $D^{2} \geq 1 + k /4$ in the dual of a finite-height segment forest and in the infinite-height coordinate predual; this should not be substituted for the main manuscript’s stated bound without identifying the spaces and norms.

The tree-based companions provide several further realizations of the separation. Four spaces defined by bounded tree potentials satisfy $δ (t) \geq 1 + t^{2} /4 - 1$ for $0 < t < 1$ and admit no AUC renorming. On finite-height trees, a globally constrained norm is identified with the least additive cost of a Hilbert vector and root paths. The quadratic path and segment-start outer Hilbert sums are reflexive examples. Another manuscript computes norms using two least nonnegative fields whose difference is the vector: with Euclidean child aggregation, two root-sum spaces satisfy $δ (t) \geq t^{3} /128$ for $0 < t < 1$ , including a reflexive joining-root space. It also gives a reflexive fixed-zero-root construction and further estimates for height-dependent aggregation.

The two $L^{1}$ companions use different settings. For an infinite-dimensional real subspace whose unit ball is totally bounded in measure and whose norm has the Daugavet property, the averaged midpoint and one-sided asymptotic moduli at every unit center are exactly $max {t /2, t - 1}$ and $max {0, t - 2}$ ; weak-neighborhood geometry excludes every equivalent AUC norm. A quantitative Kadets–Werner construction supplies both hypotheses. Separately, closed real $L^{1}$ spans of path products of independent exponential or Gaussian-square multipliers have positive averaged midpoint moduli and no equivalent AUC norm. For that construction, the renorming obstruction extends to every positive, nonconstant, mean-one multiplier with finite second moment.

Useful notions

Norms and Banach spaces
Equivalent norms and renorming
Reflexivity
Asymptotic convexity and its moduli
Graph embeddings and distortion

What was known before / what is new

Before

The repository summary attributes an earlier separation to Baudier and says that the present construction extends it to reflexive spaces. The supplied text gives no further details of that earlier result. The main abstract also frames its conclusion as a negative answer to the reflexive diamond converse for asymptotic uniform convexifiability.

New

The main contribution is a separable reflexive real example combining asymptotic midpoint uniform convexity with the impossibility of any equivalent AUC norm. It supplies explicit lower bounds for both the averaged midpoint modulus and diamond distortion. The companions broaden the constructions through segment spaces, tree potentials, recursive norms, and real $L^{1}$ spaces, with additional quantitative and exact estimates.

Limits and assumptions

The main example is a separable reflexive real Banach space; the conclusion is an existence result, not a statement that every reflexive midpoint-convex space has these obstructions. The companion estimates belong to their specified spaces and hypotheses: in particular, several tree-modulus bounds are stated only for $0 < t < 1$ , and the exact $L^{1}$ formulas require both total boundedness in measure of the unit ball and the Daugavet property. The general multiplier statement concerns the obstruction to AUC renorming, not an explicitly stated midpoint estimate for every such multiplier. The main result has a Lean formalization, and all seven manuscripts are marked Lean-formalized, but the supplied text does not give claim-by-claim coverage. It identifies no further open question beyond the converse that the main manuscript answers negatively.

Mini glossary

Banach space: A normed vector space in which every Cauchy sequence converges.
Equivalent norms: Two norms that bound one another up to positive constant factors.
Reflexivity: The property that the canonical map from a Banach space to its double dual is onto.
Averaged asymptotic midpoint modulus: A quantitative measure of asymptotic convexity based on averaged behavior around a midpoint.
AUC renorming: A choice of equivalent norm that is asymptotically uniformly convex.
Distortion: The multiplicative discrepancy between distances in a space or graph and their images under an embedding, after allowing a common rescaling.

Keywords : Asymptotic midpoint uniform convexity · Reflexive Banach spaces · Diamond distortion

AI-generated explanation based on the manuscript. Unofficial. Refer to the manuscript for the authoritative version.

3 · Technical details and sources

Claim made in the manuscript (source text)

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.

Verified by Lean: the Lean software checked the part of the reasoning described below, sometimes only part of the manuscript. This does not mean independent mathematicians have validated the result.

Scope of the Lean formalization (source text)

27/09/2026

Verified by Lean

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 $t^{3} /128$ 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.

PDF GitHub folder Lean formalization