Nonpositive curvature is not coarsely universal

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 Ec Norm Super (4) 51(3):657–700, 2018). We establish this statement by proving that a metric space which is q-barycentric for some $$q\in [1,\infty )$$ 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 $$\ell _\infty $$ grid $$[m]_\infty ^n=(\{1,\ldots ,m\}^n,\Vert \cdot \Vert _\infty )$$ into $$\ell _q$$ for all $$q\in (2,\infty )$$ , from which we deduce the following discrete converse to the fact that $$\ell _\infty ^n$$ embeds with distortion O(1) into $$\ell _q$$ for $$q=O(\log n)$$ . A rigidity theorem of Ribe (Ark Mat 14(2):237–244, 1976) implies that for every $$n\in {\mathbb {N}}$$ there exists $$m\in {\mathbb {N}}$$ such that if $$[m]_\infty ^n$$ embeds into $$\ell _q$$ 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.