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arXiv · 2605.28311

On metric characterizations of tree and fragmentability indices of Banach spaces

Estelle Basset·Gilles Lancien·Antonin Procházka
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Abstract

We introduce two ordinal indices that are linear invariants for Banach spaces: the dyadic tree index and the sprawling tree index. We show that they are also bi-Lipschitz invariants. In fact, we characterize their values in terms of sub-Lipschitz embeddability of dyadic or countably branching diamond graphs of ordinal height. We derive applications for separable Banach spaces that are universal for complete countable metric spaces and bi-Lipschitz embeddings. We also discuss the links of these tree indices with classical fragmentability indices of Banach spaces such as the dentabilty, weak fragmentability and Szlenk indices.

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Estelle Basset, Gilles Lancien, Antonin Procházka. 2026-05-27. On metric characterizations of tree and fragmentability indices of Banach spaces. https://arxiv.org/abs/2605.28311

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