1 Introduction
Asymptotic convexity describes the convexity that remains after finitely many directions are discarded. The question considered here is whether symmetric midpoint convexity can persist in a reflexive Banach space which admits no asymptotically uniformly convex equivalent norm.
All spaces and scalars are real. For an infinite-dimensional Banach space Y with norm N, write \operatorname{cof}(Y) for the closed finite-codimensional subspaces of Y. For t>0, define
The norm is asymptotically uniformly convex (AUC) if \overline\delta_N(t)>0 for all t>0, and asymptotically midpoint uniformly convex (AMUC) if \widehat\delta_N(t)>0 for all t>0. Thus AUC requires a gain in every sufficiently far-out direction; AMUC permits the two opposite directions to be tested together. A norm N is equivalent to a given norm \|\cdot\| if a\|x\|\le N(x)\le b\|x\| for some 0<a\le b<\infty and all x. A space is AUC renormable if it admits an equivalent AUC norm.
Dilworth, Kutzarova, Randrianarivony, Revalski, and Zhivkov introduced AMUC and constructed an AMUC norm on \ell_2 which is not AUC [6, Definition 2.2 and Theorem 2.4]. Their question about equivalence up to renorming [6, Section 5] has already been answered negatively by Baudier [4, Theorem 1 and Corollary 1], using the Kadets–Werner modification of the Bourgain–Rosenthal space [10, Theorem 2.5]. We give a reflexive example with an explicit midpoint estimate.
The historical definition of AMUC uses the maximum of the two signed norms in (1.2). This gives the same positivity property. The average never exceeds the maximum. Conversely, intersect any chosen finite-codimensional subspace with the kernel of a functional norming x. Both signed norms are then at least one, so a gap for their maximum gives half that gap for their average. Our quantitative modulus always uses the average in (1.2).
The example also addresses a metric question. Define the countably branching diamonds by taking D_0 to be a single edge and replacing every edge at each step by countably many internally disjoint two-edge paths. Each current edge has length one; let d_k be the shortest-path metric on D_k. An embedding of D_k into a Banach space has distortion at most D if it is a map f for which some scale s>0 satisfies
The family embeds uniformly when one finite D works for all k. For finite binary diamonds, Lee and Naor obtained quantitative L_p distortion bounds by summing convexity losses [11, Lemma 2.2], and Johnson and Schechtman developed a self-improvement proof [9, Section 4]. Baudier, Causey, Dilworth, Kutzarova, Randrianarivony, Schlumprecht, and Zhang proved that an equivalent AMUC norm prevents uniform embeddings of the countably branching diamonds [3, Theorem 4.1]. For reflexive spaces with unconditional asymptotic structure, they also proved that failure of AUC renormability implies uniform diamond embeddings [3, Theorem B and Corollary 5.3]. Swift extended the embedding construction to countably branching bundle graphs under the same reflexive, unconditional, and non-AUC-renormable hypotheses [14, Theorem 3.2 and Corollary 3.4]. Perreau subsequently treated duals of separable spaces with weak-star unconditional asymptotic structure, retaining unconditional asymptotic structure in the reflexive formulation [13, Theorems 4.8 and 5.3]. The example below shows that the diamond implication fails for general reflexive spaces.
More recently, Basset, Lancien, and Procházka related AMUC to separated families of midpoints and obtained the diamond obstruction even when the maps are controlled only on specified vertex pairs [2, Definition 6.15 and Propositions 6.16 and 6.20]. Malthaner obtained quantitative obstructions for countably branching diamonds using a Matoušek-type coloring and path argument [12, Theorems 3.5 and 3.7]. We use the full bi-Lipschitz convention (1.3) and prove the required midpoint estimate directly.
1.1 The example and result
For each integer h\ge1, let T_h=\mathbb N^{\le h} be the rooted tree of words of length at most h, ordered by extension of words. Its root is the empty word. Keep separate copies of these trees in the forest
Write a\preceq b if a is an ancestor of b in the same component. A segment is a nonempty interval [a,b]=\{v:a\preceq v\preceq b\} in an ancestral chain. For a finitely supported real function u on \mathcal F, set u(S)=\sum_{v\in S}u(v) and define
The Euclidean triangle inequality and singleton segments show that \rho is a norm, with \|u\|_2\le\rho(u)\le\|u\|_1. Let J be its completion and let X=J^*, with the dual norm denoted by \|\cdot\|. We write e_v\in J for a coordinate unit vector. Singleton tests give |u(v)|\le\rho(u), so coordinate evaluation extends to an element e_v^*\in X. The construction in (1.4) uses James’s square-sum norm on disjoint segments [8, p. 739]; the branching and component heights here differ from the classical binary tree.
For the classical binary James tree space, Girardi proved AUC of both the coordinate predual and the full dual [7, Theorems 3 and 5]. The countable branching and finite, unbounded component heights used here lead to the following different behavior.
Theorem 1.1. The space X=J^* defined by (1.4) is infinite dimensional, separable, and reflexive, and has no equivalent AUC norm. Its given norm satisfies
For every integer k\ge0, every embedding of D_k into X has distortion at least \sqrt{1+k/12}.
This gives a negative answer to the reflexive diamond question recorded as Problem 39 in [5, author manuscript p. 388]. The metric obstruction is unchanged by equivalent renorming, although the numerical constants refer to the specified dual norm.
1.2 How the argument fits together
Section 2 establishes reflexivity and the coordinate projections used throughout the proof. A sum of coordinate functionals along one root path has norm one, while its next coordinate increment is a weakly null unit vector as the child varies. In Section 3, a hypothetical AUC norm forces repeated multiplicative gains along one such path. Arbitrarily high finite trees contradict its bounded norm.
The midpoint argument in Section 4 must control segment tests crossing from a finite ancestral set (one closed under taking ancestors) to its complement. Simply adding norming tests for the two parts creates uncontrolled positive cross terms. We add one correction at their first exit vertices. The correction makes both resulting tests inexpensive, and its evaluations cancel when the two endpoint norms are averaged. Lemma 4.1 expresses the result as a lower bound in terms of the squared head and tail norms. Section 4.1 extends this estimate to coordinate completions on general rooted forests.
Finally, the infinitely many middle vertices of a diamond supply a separated family of approximate midpoints. Two of their finite coordinate projections can be made arbitrarily close. Their half-difference therefore reduces the metric problem to the head–tail estimate. Section 5 turns this into a loss in squared stretch at every replacement and sums the losses. This gives a direct quantitative instance of the midpoint and self-improvement methods for these graphs [3, Section 4].
2 The coordinate structure of the forest
We first establish the structural facts needed to separate a vector into finitely many coordinates and its complement. The finite heights give reflexivity and coordinate density; the ancestral order gives contractive projections and an exact norm formula on the complementary subtrees.
For a finite set H, the coordinate projections are P_Hu=\sum_{v\in H}u(v)e_v on J and P_Hx=\sum_{v\in H}x(e_v)e_v^* on X; write Q_H=I-P_H.
Lemma 2.1. The space X defined by (1.4) is infinite dimensional, separable, and reflexive, and the coordinate functionals have dense span in X. Let H\subset\mathcal F be finite and ancestral, meaning that it contains every ancestor of each of its vertices. The coordinate projections P_H and Q_H are contractive on both J and X. Moreover, P_Hx\to x along any increasing ancestral exhaustion of \mathcal F.
Proof. Singleton tests give \|u\|_2\le\rho(u). On a component of height h, each segment has at most h+1 vertices. Thus Cauchy–Schwarz and disjointness give
Write J_h for the completion on that component. Segment families from different components combine freely, so
Each J_h is isomorphic to Hilbert space. The dual of an \ell_2 sum is isometrically the \ell_2 sum of the duals: Cauchy–Schwarz proves one inequality, and testing on almost norming unit vectors in finitely many components, with proportional weights, proves the other. Applying this identity twice proves reflexivity. In particular X^*=J canonically. If the coordinate functionals did not have dense span in X, Hahn–Banach would give a nonzero element of X^*=J annihilating them. All its coordinates would vanish, a contradiction. Countability of \mathcal F now proves separability; the independent coordinates prove infinite dimensionality.
A segment meets H in a prefix and its complement in a suffix. Restricting disjoint segment tests to either set therefore proves contractivity on J. The adjoints give precisely the coordinate projections on X, and remain contractive. Finally, approximate x by a finite coordinate combination. Once H contains its support, contractivity bounds the discarded tail of x by that approximation error. This proves convergence along an ancestral exhaustion.
In particular, the completion of the finite coordinate functionals in
is exactly X, under the coordinate pairing. This identifies the dual and coordinate-completion descriptions of the present finite-height forest. Each component has infinitely many vertices; only its height is finite, and no Hilbert equivalence constant uniform in the height was needed.
For x=P_Hx, restriction to P_HJ preserves its functional norm. The finite-dimensional unit ball is compact, so there is f=P_Hf with \rho(f)\le1 and x(f)=\|x\|. If x\ne0 then \rho(f)=1; if x=0, including when H is empty, take f=0.
Let G_H be the set of minimal vertices outside H. We call these vertices gates. They include roots of components disjoint from H. Distinct gates are incomparable, and the descendant sets U_g=\{v:g\preceq v\} partition \mathcal F\setminus H. Each segment crossing from H to its complement passes through one gate. Figure 1 illustrates the partition.
Figure 1: A finite ancestor-closed head and some of its gates. Black large nodes belong to H; circled nodes are gates. Shading indicates two complementary descendant cones. Only finitely many of the countably many branches are drawn. In particular an untouched component begins with a gate at its root.
Let J(U_g) denote the coordinate subspace of J supported in U_g, with its inherited norm.
Lemma 2.2. For every finite ancestral H,
The gate-coordinate map x\mapsto(x(e_g))_{g\in G_H} is a contraction from X to \ell_2(G_H).
Proof. On a vector supported outside H, every segment test reduces to a segment in one cone. Grouping a disjoint family by its gates gives the upper estimate for the first identity; combining almost maximizing families from finitely many distinct cones gives the reverse estimate. Pass to the completion. A functional supported outside H has exactly the norm of its restriction to Q_HJ, because Q_H is contractive. Duality for the \ell_2 sum therefore proves the second identity.
A segment contains at most one gate, so \rho(u)=\|u\|_2 for vectors supported on G_H. Testing x on such finite vectors proves the gate-coordinate contraction.
If A is any antichain and (a_v)_{v\in A} has finite support, then
The first equality with the Euclidean norm follows because a segment meets A at most once. For the dual norm, singleton tests give the upper bound, and testing on the proportional vector supported on A gives the lower bound. Thus these identities concern the specified norm, not merely equivalent norms on the same coordinate space.
3 Why no equivalent norm is AUC
The renorming obstruction uses bounded root paths and weakly null child increments. We first isolate the elementary finite-codimensional argument; its midpoint counterpart will be proved separately.
Lemma 3.1 (Modulus conventions). For an infinite-dimensional real Banach space Y with norm N, let m_N be the modulus obtained from (1.2) by replacing the average by \max\{N(x+ty),N(x-ty)\}. Then, for t>0,
For either symmetric modulus the inner infimum is unchanged if N(y)=1 is replaced by N(y)\ge1. Both symmetric moduli, including their centerwise versions before taking the infimum over x, are nondecreasing. Positivity on (0,1) therefore implies positivity at every positive radius.
Proof. For fixed x,y, the average and maximum of the two signed norms are even convex functions of the radius, hence nondecreasing on [0,\infty). This proves the assertions about radii and monotonicity. A norming functional for a unit center x makes both signed norms at least one on its kernel. Intersecting any chosen finite-codimensional subspace with this kernel proves nonnegativity of the one-sided modulus and that the maximum excess is at most twice the average excess, after taking the supremum over subspaces. The average is pointwise at most the maximum. Finally, on a fixed subspace the infimum of the one-sided excess is a lower bound for each of the two signed excesses, since y and -y both belong to that subspace. Taking the successive suprema and infima proves the remaining comparison.
Lemma 3.2 (Bounded trees). Let Y be a real Banach space. Suppose that for every integer h\ge1 there are vectors (u_s^{(h)})_{s\in\mathbb N^{\le h}} and constants m,M,c>0, independent of h, such that
For every nonterminal s, suppose the child increments tend weakly to zero as j\to\infty. Then every equivalent norm satisfying \alpha\|\cdot\|\le N\le\beta\|\cdot\|, with 0<\alpha\le\beta<\infty, obeys
The same conclusion holds for one infinite tree with these properties.
Proof. Put t_0=\alpha c/(2\beta M). If the displayed modulus is positive, choose 0<\gamma<\overline\delta_N(t_0). At a current vector u=u_s^{(h)}, the definition supplies a finite-codimensional F such that
Convexity extends this to every w\in F with N(w)\ge t_0N(u): for a unit center z and r\ge t_0,
The weakly null increments d_j=u_{s^\frown j}^{(h)}-u have images tending to zero in norm in the finite-dimensional quotient Y/F. Thus there are w_j\in F with N(w_j-d_j)\to0. Eventually N(w_j)\ge\alpha c/2\ge t_0N(u). Taking the approximation error at most \gamma N(u)/2 gives a child with
The gain is uniform although F may depend on the current node. Choose h with \alpha m(1+\gamma/2)^h>\beta M and make this choice successively along a branch of that tree. The resulting norm exceeds \beta M, a contradiction. The increments in particular force Y to be infinite dimensional, and Lemma 3.1 gives the reverse nonnegative inequality in (3.1). An infinite tree may be restricted to any sufficiently large height.
We apply the lemma to the forest. For a node v in the component with root r, put p_v=\sum_{r\preceq w\preceq v}e_w^*. The root path is one segment test, so \|p_v\|\le1; evaluation at a coordinate on that path gives equality. At a nonterminal node the child increments are e_{v^\frown j}^*, each of norm one. They tend weakly to zero: X^*=J, and every u\in J has square-summable coordinates, so u(v^\frown j)\to0. We therefore have m=M=c=1 in Lemma 3.2. For every equivalent norm \alpha\|\cdot\|\le N\le\beta\|\cdot\|,
This proves the renorming assertion in Theorem 1.1.
4 A paired extension of segment tests
The renorming obstruction is proved. To establish midpoint convexity, we now show that a symmetric pair must grow when its center and displacement occupy a finite ancestral set and its complement. The squared-norm form will also be the input for the diamond argument.
Lemma 4.1. Let H\subset\mathcal F be finite and ancestral. If x,y\in X satisfy x=P_Hx and y=Q_Hy, then
More generally, for all s,t>0,
Proof. We first prove, for every s>0,
Suppose initially that y has finite support. Choose finite tests f,k\in J, supported in H and its complement respectively, with \rho(f),\rho(k)\le1 and pairings as close as desired to \|x\| and \|y\|. The naive tests f+sk and f-sk can have large positive crossing terms. We will add the same vector to both tail tests; its contribution to the endpoint average will cancel. To control its size, at each gate g put
Only finitely many m_g are nonzero, and \sum_gm_g^2\le1: choose an almost maximizing initial segment in each of finitely many distinct cones and use their disjointness.
For a gate with parent v\in H, consider the suffixes [a,v] whose f-sum has magnitude at least 2sm_g. If m_g>0 and such suffixes exist, choose the shortest one and let \sigma_g be opposite to the sign of its sum. In all other cases put \sigma_g=1. Set
The gates form an antichain, so \rho(c)\le1 and \rho(v^\pm)\le2. Every initial segment in the cone of g has v^\pm-sum of sign \sigma_g or zero, and magnitude at most 2m_g.
We claim \rho(f+sv^\pm)^2\le1+12s^2. Fix a disjoint family of segments and denote their f- and v^\pm-sums by \alpha_i and \beta_i. We have \sum_i\beta_i^2\le4. A positive cross term \alpha_i\beta_i can occur only at a crossing segment, with some gate g; different crossing segments use different gates. If |\alpha_i|\le2sm_g, then 2s\alpha_i\beta_i\le8s^2m_g^2. Otherwise the head itself reaches the threshold, so it contains the shortest selected suffix. Write \alpha'_i for that suffix’s f-sum. Since \alpha_i\beta_i>0 and \beta_i has sign \sigma_g, the choice of \sigma_g gives \alpha_i\alpha'_i<0. Thus the selected suffix is proper, and |\alpha'_i|\ge2sm_g by its selection. Splitting it off the head increases the sum of squared f-sums by
Figure 2 shows the proper suffix and the cut, with the segment index suppressed.
Figure 2: The threshold cut for a crossing segment. Its head has f-sum \alpha and its selected suffix has f-sum \alpha'. When \alpha\beta>0 and |\alpha|>2sm_g, the sign choice gives \alpha\alpha'<0 and |\alpha'|\ge2sm_g. Cutting immediately before a' therefore increases the squared f-test by at least 2s\alpha\beta. The vertex spacing is schematic.
These splits involve distinct original segments, so all can be made simultaneously. Their pieces and the remaining heads are still disjoint. Thus \sum_i\alpha_i^2 together with all the large positive cross terms is at most \rho(f)^2\le1. The small cross terms cost at most 8s^2, and the pure tail terms cost at most 4s^2. We have controlled both corrected tests. It remains to use them together so that their common correction disappears.
Test x+y against (f+sv^+)/\sqrt{1+12s^2} and x-y against (f+sv^-)/\sqrt{1+12s^2}. The mixed support terms vanish, and the two evaluations involving c cancel in the average. Letting the norming errors tend to zero proves (4.3). Finite tails are dense in Q_HX, so (4.3) holds for every y=Q_Hy. If \|x\|>0, maximize its right side by taking s=\|y\|/(12\|x\|); zero values follow by a limit. The maximum is the right side of (4.1). Replacing y by ty in (4.3) proves (4.2).
Proof of the midpoint assertion in Theorem 1.1. For a unit x\in X, choose H with \|x-P_Hx\|<\eta. For every unit y\in\ker P_H, Lemma 4.1 applied to P_Hx and ty gives
Take the infimum over these y, the supremum over finite-codimensional subspaces, and let \eta\downarrow0. This proves the asserted bound, since it is independent of the unit center x.
4.1 The coordinate completion on a general forest
The paired argument used finite support and ancestral cuts, rather than finite component heights. We record its resulting scope, while keeping the distinction between a coordinate completion and a full dual explicit. Let T be any countably infinite rooted forest in which every vertex has finitely many ancestors. Define \rho_T by (1.4) with T in place of \mathcal F, let J(T) be the completion, and put
Thus X(T) is the completion of finite coordinate vectors in the polar norm. We do not identify it with all of J(T)^*.
A finite ancestral set cuts each segment into a prefix and suffix. The projections P_H,Q_H are therefore contractive on J(T) and X(T), and finite vectors outside H are dense in Q_HX(T). Every finite set has finite ancestral closure, so an increasing finite ancestral exhaustion gives P_Hx\to x for all x\in X(T). The proof of Lemma 4.1 applies unchanged to these objects: there are finitely many nonzero masses m_g, their initial segments are disjoint, and unattached roots give no crossing term. These are all the properties used in the test construction, the simultaneous suffix cuts, and the passage to completed tails. Consequently (4.1) and (4.2) hold in X(T), and the same arbitrary-center approximation gives
This also explains why reflexivity is used for the example’s structural identification, rather than as a hypothesis of the paired estimate.
For T=\mathbb N^{<\omega}, the coordinate completion X(T) is the countably branching James-tree predual B_\infty, as described in Banakh [1, §2.II] following Ghoussoub–Maurey–Schachermayer. We give its path obstruction in the displayed norm. Each root-path functional p_v=\sum_{w\preceq v}e_w^* again has norm one. The antichain calculation in (2.2) uses only disjoint segments and applies to X(T) as well. Thus, for any \phi\in X(T)^* and finitely many children of a fixed vertex,
Taking a_j=\phi(e_{v^\frown j}^*) shows that these evaluations are square summable, and hence tend to zero. The child increments are therefore weakly null against the whole continuous dual. Applying Lemma 3.2 with m=M=c=1 proves \overline\delta_N(\alpha/(2\beta))=0 for every equivalent norm \alpha\|\cdot\|\le N\le\beta\|\cdot\| on this coordinate completion. This argument does not require it to be reflexive.
5 From approximate midpoints to diamonds
5.1 An infinite family of approximate midpoints
Support restrictions disappear when infinitely many separated perturbations are available. This is the only compactness step in the metric proof. Approximate-midpoint lenses are central to AMUC [6, Theorem 2.1]; the broader midpoint method is attributed to Enflo’s unpublished argument in [3, Section 4]. The configuration below is a separated midpoint family, called a spider in [2, Definition 3.7; Definition 6.15 and Proposition 6.16]. We derive its precise squared deficit from Lemma 4.1.
Lemma 5.1. Let R>0, \varepsilon>0, x\in X, and (y_j)_{j\ge1}\subset X satisfy
Then
Proof. Convexity gives \|x\|,\|y_j\|\le R. Given \eta>0, choose finite ancestral H with \|Q_Hx\|<\eta. The bounded sequence (P_Hy_j) is in a finite-dimensional space, so some distinct i,j satisfy \|P_H(y_i-y_j)\|<2\eta. Put z=(y_i-y_j)/2. Since
we have \|x\pm z\|\le R. Also
Both P_Hx\pm Q_Hz have norm at most R+2\eta. By Lemma 4.1,
For 0<\eta<\varepsilon/2, let \eta\downarrow0 to obtain (5.1).
5.2 Adding the loss over the diamond levels
Figure 3 shows the replacement whose loss we now estimate.
Figure 3: One diamond replacement, with only three of the countably many paths shown. All new edges have length one. The old endpoints are at distance two, as are any two distinct new middle vertices.
Each parent edge will consume the same positive amount of squared stretch. Earlier quantitative diamond arguments accumulate convexity losses across levels [11, Lemma 2.2], or use a self-improvement induction [9, Lemma 1 and Proposition 2]. The countably branching AMUC form appears in [3, Section 4]. Here Lemma 5.1 supplies a fixed squared loss, which we can add directly.
Proof of the distortion assertion in Theorem 1.1.
Let f:D_k\to X satisfy s d_k\le\|f(u)-f(v)\|\le Ds d_k. Divide by s, so f is noncontracting and D-Lipschitz. Distances between old vertices double at each refinement: a new path between old vertices consists of two-edge passages which either return to their starting vertex or contract to an old edge, and every old path can be lifted with twice its length. Therefore the endpoints of an edge of D_m have distance 2^{k-m} in D_k. Distinct middle vertices replacing that edge also have distance 2^{k-m} in D_k.
Put
These suprema need not be attained; all that is used is 1\le L_m\le D. Fix one parent edge at level m<k, put \ell=2^{k-m-1}, and let w_j be its middle vertices. Set
The child edges give \|x\pm y_j\|\le L_{m+1}, and the middle-vertex distances give \|y_i-y_j\|\ge2. Applying Lemma 5.1 and then taking the supremum over the parent edges yields
Summing over m gives
For k=0 this is just D\ge1. This completes the proof of Theorem 1.1.
To verify the renorming assertion in the introduction, suppose a\|z\|\le N(z)\le b\|z\|. An embedding with distortion at most D for N has distortion at most bD/a for \|\cdot\|. Thus uniform embeddings for an equivalent norm would contradict the distortion bound just proved.
References
- [1]Taras Banakh. The topological classification of weak unit balls of Banach spaces. Dissertationes Mathematicae, 387:7–35, 2000. https://www.impan.pl/shop/publication/transaction/download/product/88099.
- [2]Estelle Basset, Gilles Lancien, and Antonín Procházka. On metric characterizations of tree and fragmentability indices of Banach spaces. 2026. Version 1, 27 May 2026. doi:10.48550/arXiv.2605.28311, arXiv:2605.28311.
- [3]Florent Baudier, Ryan Causey, Stephen Dilworth, Denka Kutzarova, Nirina L. Randrianarivony, Thomas Schlumprecht, and Sheng Zhang. On the geometry of the countably branching diamond graphs. Journal of Functional Analysis, 273(10):3150–3199, 2017. Theorem and page locators refer to arXiv:1612.01984v1 (2016). doi:10.1016/j.jfa.2017.05.013, arXiv:1612.01984.
- [4]Florent P. Baudier. The Kadets–Werner modification of Bourgain–Rosenthal space is asymptotically midpoint uniformly convex. 2026. Version 1, 18 September 2026. doi:10.48550/arXiv.2609.21283, arXiv:2609.21283.
- [5]Florent P. Baudier and Gilles Lancien. Asymptotic and nonlinear geometries of Banach spaces and their interactions. Cours Spécialisés 33, Société Mathématique de France, 2026. Problem and page locators refer to arXiv:2512.00817v1, 30 November 2025. arXiv:2512.00817, https://arxiv.org/abs/2512.00817v1.
- [6]S. J. Dilworth, Denka Kutzarova, N. Lovasoa Randrianarivony, J. P. Revalski, and N. V. Zhivkov. Lenses and asymptotic midpoint uniform convexity. Journal of Mathematical Analysis and Applications, 436(2):810–821, 2016. Author-manuscript locators refer to the version dated 8 October 2015. doi:10.1016/j.jmaa.2015.11.061.
- [7]Maria Girardi. The dual of the James tree space is asymptotically uniformly convex. Studia Mathematica, 147(2):119–130, 2001. doi:10.4064/sm147-2-2, arXiv:math/0004166.
- [8]Robert C. James. A separable somewhat reflexive Banach space with nonseparable dual. Bulletin of the American Mathematical Society, 80(4):738–743, 1974. doi:10.1090/S0002-9904-1974-13580-9.
- [9]William B. Johnson and Gideon Schechtman. Diamond graphs and super-reflexivity. Journal of Topology and Analysis, 1(2):177–189, 2009. doi:10.1142/S1793525309000114.
- [10]Vladimir Kadets and Dirk Werner. A Banach space with the Schur and the Daugavet property. Proceedings of the American Mathematical Society, 132(6):1765–1773, 2004. Preprint version: arXiv:math/0301182v1, 17 January 2003. doi:10.1090/S0002-9939-03-07278-2, arXiv:math/0301182.
- [11]James R. Lee and Assaf Naor. Embedding the diamond graph in L_p and dimension reduction in L_1. Geometric and Functional Analysis, 14(4):745–747, 2004. doi:10.1007/s00039-004-0473-8.
- [12]Ryan Malthaner. On Matoušek-like embedding obstructions of countably branching graphs. Journal of Mathematical Analysis and Applications, 543(2):128896, 2025. Part 1. Preprint theorem locators refer to v2, 8 October 2023; journal numbering differs. doi:10.1016/j.jmaa.2024.128896, arXiv:2310.03257.
- [13]Yoël Perreau. On the embeddability of countably branching bundle graphs into dual spaces. 2021. Version 1, 21 April 2021. doi:10.48550/arXiv.2104.10494, arXiv:2104.10494.
- [14]Andrew Swift. A coding of bundle graphs and their embeddings into Banach spaces. Mathematika, 64(3):847–874, 2018. Theorem and page locators refer to arXiv:1710.00877v1 (2017). doi:10.1112/S002557931800027X, arXiv:1710.00877.