Reflexive midpoint convexity and diamond distortion

The explanation

Take the unit vectors x = (1, 0) and y = (0, 1) in the plane with ordinary length. The vectors x + y and x − y both have length √2, about 1.414. Their average length exceeds 1 by about 0.414. That gain is a sign the unit ball is round. Now use the max length, the larger of the two coordinates. Then x + y = (1, 1) has length exactly 1. There is no gain, because the unit ball is a square with flat sides.

A Banach space is a space of vectors with a length rule, a norm, where limits behave well. In infinite dimensions, asymptotic convexity asks for such gains when y is pushed far out. Far out means y avoids any finite set of directions you name. The norm is asymptotically uniformly convex, or AUC, if x + ty always gains length. It is asymptotically midpoint uniformly convex, or AMUC, if the average of x + ty and x − ty gains. AMUC was introduced in 2016, with a norm on the sequence space ℓ2 that is AMUC but not AUCSource 1. Baudier then showed that a space can have an AMUC norm but no equivalent AUC normSource 2. An equivalent norm changes lengths by at most fixed factors. His example was not reflexive. Reflexive means the space equals its double dual, the space of linear measurements of its linear measurements.

The question also has a metric side. A countably branching diamond starts as one edge. At each step, every edge becomes countably many two-edge paths. Distortion measures how much an embedding must stretch some distances relative to others. An AMUC norm stops diamonds from embedding with one bound on distortionSource 3. For reflexive spaces with extra structure, failing to be AUC renormable gives such embeddings. The open problem was whether that holds for every reflexive spaceSource 4.

The main paper says no. It builds a separable reflexive space from norms on segments of a tree. The space has no equivalent AUC normSource 5. Its midpoint gain at step t is at least √(1 + t²/12) − 1. At t = 1, that is about 0.0408. Every embedding of the depth-k diamond has distortion at least √(1 + k/12)Source 6. For k = 12, that is √2, about 1.414. For k = 96, it is 3. For k = 1,200, it is about 10.05. So no single bound works for all depths. The obstacle was controlling test functions that cross between a finite set of tree vertices and the rest. The paper adds one correction term at the exit verticesSource 7.

The six companion papers build related examples. One proves the sharper bound D² ≥ 1 + k/4 for a closely related spaceSource 8. Another computes the exact moduli for a special subspace of L1, the space of integrable functions. They are max{t/2, t − 1} and max{0, t − 2}Source 9.

How sure is this? Lean, a computer proof checker, has formal statements of the main results of all seven papers. For the main space, Lean covers reflexivity, the midpoint bound, the absence of AUC norms and the diamond bound. Some auxiliary deductions in the diamond paper assume companion inputs. So the papers prove their main results, as far as the Lean statements match. No specialist review is published.

What it doesn't show. This does not change the known positive results for reflexive spaces with unconditional asymptotic structure; the example lies outside that class.