Midpoint convexity without asymptotically convex renormings

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Family 331 · Functional analysis
Importance 5/5 · Provisional editorial rating
Sources consulted 2026-10-25

1. The question

Convexity says that the midpoint of two points in a ball remains in the ball. Stronger forms ask how far inside it must lie. In infinite-dimensional spaces, one can also ignore finitely many directions and study the geometry that remains.

Two versions of this asymptotic question look similar. One asks for growth in every sufficiently far-out direction from a unit vector. The other allows a direction and its opposite to be tested together, through the average of the two endpoint norms. Does this weaker midpoint growth imply that some equivalent norm has the stronger one-sided property?

The general implication already had a negative answer before this family. The seven manuscripts develop more explicit examples, exact moduli, and quantitative metric consequences. A central claim is a reflexive counterexample: midpoint growth persists, no equivalent asymptotically uniformly convex norm exists, and increasingly deep countably branching diamond graphs require unbounded distortion to embed.

2. Background and essential definitions

Let $X$ be an infinite-dimensional real Banach space with norm $N$. A closed subspace has finite codimension when its quotient space has finite dimension. Restricting perturbations to such a subspace discards finitely many linear measurements.

For a unit vector $x$, first choose a finite-codimensional subspace $F$, then consider every unit vector $y\in F$. The one-sided and averaged midpoint moduli are

$$\overline\delta_N(t) =\inf_{N(x)=1}\sup_F\inf_{\substack{y\in F\\N(y)=1}} \bigl(N(x+ty)-1\bigr),$$

$$\widehat\delta_N(t) =\inf_{N(x)=1}\sup_F\inf_{\substack{y\in F\\N(y)=1}} \left(\frac{N(x+ty)+N(x-ty)}2-1\right).$$

The space is asymptotically uniformly convex, AUC, when the first is positive for every $t>0$. It is asymptotically midpoint uniformly convex, AMUC, when the second is positive for every $t>0$.

The historical AMUC definition uses the maximum of the two endpoint norms. It has the same positivity property, but different numerical constants. All midpoint formulas below use the average. Some manuscripts allow directions of norm at least one; even convexity along each line gives the same infimum.

An equivalent norm changes lengths by at most fixed multiplicative factors. Saying that a space has no equivalent AUC norm is stronger than saying that its displayed norm is not AUC.

A countably branching diamond begins with one edge and repeatedly replaces each edge by countably many two-edge paths sharing the endpoints. An embedding has distortion $D$ if, after one overall rescaling, every distance is preserved between factors one and $D$. Uniform embedding means that one finite distortion works for all depths.

3. Development history and previous partial results

Dilworth, Kutzarova, Randrianarivony, Revalski, and Zhivkov introduced AMUC and exhibited a norm on $\ell_2$ that is AMUC but not AUC. Since $\ell_2$ has its usual uniformly convex norm, this did not settle the question about equivalent renormings.

Baudier subsequently answered the general renorming question negatively using the Kadets–Werner modification of the Bourgain–Rosenthal space. The example is a real $L_1$ subspace with special compactness in measure and the Daugavet property. That earlier negative answer is credited throughout this family; the present claims are refinements and different constructions, including reflexive examples.

Diamond graphs supplied a related metric question. Earlier work showed that an equivalent AMUC norm prevents uniform embeddings of countably branching diamonds. Conversely, for reflexive spaces with unconditional asymptotic structure, failure of AUC renormability implies uniform diamond embeddings. This additional structure controls the effect of signs in asymptotic configurations. Swift and Perreau developed related embedding and dual-space formulations while retaining corresponding structural hypotheses.

The proposed reflexive examples show why the unconditional assumption cannot simply be omitted from that converse. They fail AUC renormability but still obstruct uniform diamond embeddings.

Tree spaces have an older role in Banach-space geometry. James’s norm sums squared coefficients over disjoint segments of a tree. Girardi proved AUC for the canonical predual and full dual of the classical binary James tree space. These manuscripts use different branching, completions, constraints, and root conventions; the earlier binary-tree theorem does not automatically describe them.

4. What the new manuscripts claim

Segment forests and diamond distortion

M1 uses a forest containing a separate countably branching tree of each finite height. For finitely supported real arrays $u$, define

$$\rho(u)=\sup_{S_1,\ldots,S_m} \left(\sum_{i=1}^{m}\left|\sum_{v\in S_i}u(v)\right|^2\right)^{1/2},$$

where the $S_i$ are pairwise disjoint segments along ancestral chains. Complete this normed space to obtain $J$, then take its dual $X=J^*$.

M1 claims that $X$ is separable and reflexive, has no equivalent AUC norm, and satisfies

$$\widehat\delta_X(t)\geq\sqrt{1+t^2/12}-1.$$

It also gives the distortion bound $D^2\geq1+k/12$ for a depth-$k$ diamond.

M2 studies both this forest dual and a separate infinite-height coordinate space $B_\infty$, the closed span of coordinate evaluations in the dual of the infinite-tree segment space. For either model, it claims the following lens estimate. If a center $x$ is supported on a finite ancestral set $H$, and $Q_H$ removes those coordinates, then

$$\|x+y\|,\|x-y\|\leq R \quad\Longrightarrow\quad \|Q_Hy\|\leq2\sqrt{R^2-\|x\|^2}.$$

The same head $H$ works for every displacement in the lens, including displacements with a nonzero head. Both specified norms are AMUC and neither space admits an AUC renorming. Reflexivity is asserted for the finite-height forest dual.

Using this stronger lens estimate, M3 obtains

$$D^2\geq1+k/4$$

for diamonds in either of these two spaces. Its normalized edge lengths differ from M1’s convention only by an overall scale at each depth, which does not change distortion. M3 also develops direct tree-energy arguments and power-type bounds for other norms in the family.

Exact curves in an L1 model

M4 assumes that $E$ is an infinite-dimensional closed real subspace of a probability $L_1$ space, its unit ball is totally bounded for convergence in measure, and its inherited norm has the Daugavet property:

$$\|I+T\|=1+\|T\|$$

for every rank-one bounded operator $T$. Total boundedness in measure means that finitely many measure-metric balls cover the unit ball at each positive resolution.

Under these hypotheses, M4 claims exact formulas, valid at every unit center and hence for the moduli:

$$\widehat\delta_E(t)=\max\{t/2,t-1\}, \qquad \overline\delta_E(t)=\max\{0,t-2\}.$$

It separately excludes every equivalent AUC norm and supplies a quantitative realization of the earlier Kadets–Werner construction satisfying the assumptions. The formulas describe the inherited norm, not all equivalent norms.

Further explicit constructions

M5 constructs four tree norms whose scalar tests have bounded sums along root paths and different quadratic constraints: on sibling groups, antichains, or the whole coefficient family. An antichain consists of mutually incomparable vertices. All four have

$$\widehat\delta(t)\geq\sqrt{1+t^2/4}-1 \qquad(0<t<1)$$

and admit no AUC renorming. Further path-cost and segment-start constructions yield reflexive AMUC examples with the same renorming obstruction. These are distinct norms; the constants are not interchangeable.

M6 gives a probabilistic construction. Put independent copies of a positive, nonconstant, mean-one random variable $W$ at the nonroot vertices of a countably branching tree. Multiply along each root path, then take the closed real $L_1$ span of the resulting products. For every such $W$ with finite second moment, M6 excludes an equivalent AUC norm. Positive averaged midpoint bounds are proved specifically for exponential multipliers and squares of standard Gaussian variables. It makes no claim that all multiplier laws give AMUC, or that these product spaces are reflexive or Daugavet spaces.

M7 defines norms through two least nonnegative fields on a tree whose difference equals the vector and whose values dominate the Euclidean size of their children. Root sums give an infinite-tree model and a reflexive joining-root model with

$$\widehat\delta(t)\geq t^3/128 \qquad(0<t<1).$$

A separate fixed-zero-root normalization gives a homogeneous cubic estimate. Height-dependent aggregation exponents yield further sixth-power and cubic bounds. The stated models again admit no equivalent AUC norm.

5. Why the advance matters

Importance: 5/5. The main significance is the claimed separation within reflexive spaces and its consequence for a natural metric characterization. Failure of AUC renormability would no longer force uniform diamond embeddings without additional structural assumptions.

The family also gives unusually concrete ways to see why two-sided and one-sided asymptotic behavior differ. The exact $L_1$ curves show positive averaged growth at every positive radius while one-sided growth remains zero through radius two. The tree examples turn a qualitative separation into estimates that accumulate over graph depth.

These claims refine an already established general negative answer rather than replacing its history. Their value lies in the reflexive setting, explicit constructions, exact or quantitative geometry, and different mechanisms that lead to the separation.

6. The main idea

A recurring obstruction to AUC renorming is a bounded tree of path vectors. A root-path sum or product has uniformly bounded norm, while moving to a new child adds a weakly null increment with norm bounded away from zero. Weakly null means that every bounded linear measurement tends to zero along the child sequence.

A hypothetical AUC norm would allow the successive choice of children giving a definite multiplicative increase at each step. Arbitrarily long paths would then have unbounded norm. Equivalence with the original norm contradicts the bounded path construction.

Midpoint growth requires a different argument. In the segment models, tests crossing from a finite head into a tail can have troublesome mixed terms. M1 adds corrections at first-exit vertices so their evaluations cancel between the two endpoints. M2 instead uses atomic segment representations and controls cancellations where segments leave the head. The resulting squared-radius budget bounds the whole tail of a symmetric lens.

Infinite branching connects this estimate to diamonds. Among infinitely many midpoint images, two finite-dimensional projections can be made very close. Their difference therefore retains substantial tail size. The lens estimate forces a loss in squared stretch at each replacement level; summing the losses gives M3’s depth-dependent bound.

The other models implement related ideas with different tools. M5 directs scalar potential perturbations inward so they stay within the allowed interval while sharing a quadratic budget. M7 propagates a defect through a recursion where at least one child inequality is tight, avoiding a loss that multiplies at every level.

In $L_1$, the real scalar identity

$$\frac{|a+b|+|a-b|}{2}=\max\{|a|,|b|\}$$

turns midpoint gain into mass outside an overlap region. M4 combines clipping and finite sign tests with the weak-neighborhood geometry supplied by the Daugavet property. M6 conditions on a finite subtree; fresh independent multipliers beyond its first exits force positive excess uniformly over the remaining directions. The arguments have a shared theme but do not identify the underlying spaces.

7. Scope, limitations, and open questions

All constructions here use real scalars. Numerical formulas refer to the specified averaged modulus and specified norms. Maximum-based moduli have the same positivity property but need not have the same values.

Reflexivity belongs to particular forest and joining-root models. It should not be transferred to every space in the seven manuscripts, especially the independent-product examples. Likewise, the exact $L_1$ formulas require both measure-total-boundedness and the Daugavet property.

The diamond bounds concern countably branching graphs, which have infinitely many vertices even at finite depth. They are not automatically the same statement for finite binary diamonds. The distortion convention controls every pair of vertices.

Most constants are presented as quantitative bounds, not optimal values. The exact curves in M4 are a separate stronger assertion. Sharp constants, relations among the different constructions, and converse embedding theorems under other structural hypotheses remain further questions. The manuscript claims are explained without proof or novelty verification.

8. Sources and further reading

• Baudier, the earlier negative answer to the general renorming question.
• Baudier and collaborators, On the geometry of the countably branching diamond graphs, for the midpoint obstruction and conditional converse.
• Kadets and Werner, A Banach space with the Schur and the Daugavet property, for the earlier $L_1$ construction.

Source manuscripts

• M1. OpenAI, Asymptotic midpoint uniform convexity and unbounded diamond distortion in a reflexive tree space (September 27, 2026).
• M2. OpenAI, Midpoint lenses in segment spaces (September 27, 2026).
• M3. OpenAI, Distortion of countably branching diamonds from midpoint and tree energies (September 27, 2026).
• M4. OpenAI, Exact asymptotic moduli in a Daugavet subspace of L1 (September 27, 2026).
• M5. OpenAI, Midpoint convexity from bounded tree potentials and path costs (September 27, 2026).
• M6. OpenAI, Independent products in real L1: asymptotic midpoint convexity without AUC renormings (September 27, 2026).
• M7. OpenAI, Midpoint convexity from two recursive potentials (September 27, 2026).

Sources consulted on 25 October 2026 at repository revision 1581d4c. Download the manuscript citations.