Nonpositive curvature is not coarsely universal

Alexandros Eskenazis, Manor Mendel, Assaf Naor

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9 Scopus citations


We prove that not every metric space embeds coarsely into an Alexandrov space of nonpositive curvature. This answers a question of Gromov (Geometric group theory, Cambridge University Press, Cambridge, 1993) and is in contrast to the fact that any metric space embeds coarsely into an Alexandrov space of nonnegative curvature, as shown by Andoni et al. (Ann Sci Éc Norm Supér (4) 51(3):657–700, 2018). We establish this statement by proving that a metric space which is q-barycentric for some q∈ [1 , ∞) has metric cotype q with sharp scaling parameter. Our proof utilizes nonlinear (metric space-valued) martingale inequalities and yields sharp bounds even for some classical Banach spaces. This allows us to evaluate the bi-Lipschitz distortion of the ℓ grid [m]∞n=({1,…,m}n,‖·‖∞) into ℓq for all q∈ (2 , ∞) , from which we deduce the following discrete converse to the fact that ℓ∞n embeds with distortion O(1) into ℓq for q= O(log n). A rigidity theorem of Ribe (Ark Mat 14(2):237–244, 1976) implies that for every n∈ N there exists m∈ N such that if [m]∞n embeds into ℓq with distortion O(1), then q is necessarily at least a universal constant multiple of log n. Ribe’s theorem does not give an explicit upper bound on this m, but by the work of Bourgain (Geometrical aspects of functional analysis (1985/86), Springer, Berlin, 1987) it suffices to take m= n, and this was the previously best-known estimate for m. We show that the above discretization statement actually holds when m is a universal constant.

Original languageEnglish (US)
Pages (from-to)833-886
Number of pages54
JournalInventiones Mathematicae
Issue number3
StatePublished - Sep 1 2019

All Science Journal Classification (ASJC) codes

  • General Mathematics


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