TY - JOUR

T1 - Nonpositive curvature is not coarsely universal

AU - Eskenazis, Alexandros

AU - Mendel, Manor

AU - Naor, Assaf

N1 - Publisher Copyright:
© 2019, Springer-Verlag GmbH Germany, part of Springer Nature.

PY - 2019/9/1

Y1 - 2019/9/1

N2 - 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.

AB - 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.

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U2 - 10.1007/s00222-019-00878-1

DO - 10.1007/s00222-019-00878-1

M3 - Article

AN - SCOPUS:85064281256

VL - 217

SP - 833

EP - 886

JO - Inventiones Mathematicae

JF - Inventiones Mathematicae

SN - 0020-9910

IS - 3

ER -