A separable reflexive real Banach space has an asymptotically midpoint uniformly convex given norm with averaged midpoint modulus at least √(1+t²/12)−1 for t>0, but admits no equivalent asymptotically uniformly convex norm; depth-k countably branching diamonds have distortion at least √(1+k/12) in it.
A separable reflexive real Banach space has an asymptotically midpoint uniformly convex given norm with averaged midpoint modulus at least √(1+t²/12)−1 for t>0, but admits no equivalent asymptotically uniformly convex norm; depth-k countably branching diamonds have distortion at least √(1+k/12) in it.
A separable reflexive real Banach space has an asymptotically midpoint uniformly convex given norm with averaged midpoint modulus at least √(1+t²/12)−1 for t>0, but admits no equivalent asymptotically uniformly convex norm; depth-k countably branching diamonds have distortion at least √(1+k/12) in it.
If correct, strengthens the earlier negative result to a quantitative reflexive example and separates the corresponding metric diamond criterion.
Can a reflexive midpoint-convex Banach space resist every asymptotically uniformly convex renorming?
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OpenAI reports a reflexive midpoint-convexity counterexample
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