Related papers: The Andoni-Naor-Neiman inequalities and isometric …
Gromov (2001) and Sturm (2003) proved that any four points in a $\mathrm{CAT}(0)$ space satisfy a certain family of inequalities. We call those inequalities the $\boxtimes$-inequalities, following the notation used by Gromov. In this paper,…
We establish a family of inequalities that hold true on any $6$ points in any $\mathrm{CAT}(0)$ space. We prove that the validity of these inequalities does not follow from any properties of $5$-point subsets of $\mathrm{CAT}(0)$ spaces. In…
We show that countable metric spaces always have quantum isometry groups, thus extending the class of metric spaces known to possess such universal quantum-group actions. Motivated by this existence problem we define and study the notion of…
We show that compact Riemannian manifolds, regarded as metric spaces with their global geodesic distance, cannot contain a number of rigid structures such as (a) arbitrarily large regular simplices or (b) arbitrarily long sequences of…
In this paper known results of symmetric orthogonality, as introduced by G. Birkhoff, and non-expansive nearest point projections are extended from the linear to the metric setting. If the space has non-positive curvature in the sense…
An embeddability criterion for zero-dimensional metrizable topological spaces in zero-dimensional metrizable topological groups is given. A space which can be embedded as a closed subspace in a zero-dimensional metrizable group but is not…
In this paper we prove an isoperimetric inequality of euclidean type for complete metric spaces admitting a cone-type inequality. These include all Banach spaces and all complete, simply-connected metric spaces of non-positive curvature in…
We investigate isoperimetric inequalities for Lipschitz 2-spheres in CAT(0) spaces, proving bounds on the volume of efficient null-homotopies. In one dimension lower, it is known that a quadratic inequality with a constant smaller than…
In [The Space of Spaces: Curvature Bounds and Gradient Flows on the Space of Metric Measure Spaces. Memoirs of the American Mathematical Society. American Mathematical Society, 2023], Sturm studied the space of all metric measure spaces up…
We announce results on the structure of CAT(0) groups, CAT(0) lattices and of the underlying spaces. Our statements rely notably on a general study of the full isometry groups of proper CAT(0) spaces. Classical statements about Hadamard…
In the present paper we characterize the surjective isometries of the space of compact, convex subsets of proper, geodesically complete CAT(0)-spaces in which geodesics do not split, endowed with the Hausdorff metric. Moreover, an analogue…
We show that if a proper, geodesically complete, CAT(0) homology manifold is quasi-isometric to the Euclidean space R^n then it is homeomorphic to R^n. On the other hand, we show that there exist proper, geodesically complete, CAT(0) spaces…
The negative type inequalities of a metric space are closely tied to embeddability. A result by Gupta, Newman, and Rabinovich implies that if a metric graph $G$ does not contain a theta submetric as an embedding, then $G$ has negative type.…
In this paper, we prove results concerning the large scale geometry of connected, simply connected nonabelian nilpotent Lie groups equipped with left invariant Riemannian metrics. Precisely, we prove that there do not exist quasi-isometric…
In this paper, we present a geometric condition for a family of CAT(0) spaces, which ensures that the Izeki-Nayatani invariants of spaces in the family are uniformly bounded from above by a constant strictly less than 1. Each element of…
We investigate CAT(0) metric spaces whose associated Tits boundary is compact. Prominent examples of such spaces are of course the euclidean ones. However there exist non trivial geodesically complete CAT(0) spaces with compact Tits…
Given a fixed closed manifold M, we exhibit an explicit formula for the distance function of the canonical L^2 Riemannian metric on the manifold of all smooth Riemannian metrics on M. Additionally, we examine the (metric) completion of the…
In this paper, we provide an infinite metric space $M$ such that the set $\mbox{SNA}(M)$ of strongly norm-attaining Lipschitz functions does not contain a subspace which is isometric to $c_0$. This answers a question posed by Antonio…
We prove that an open manifold $M$ of dimension at least $5$ which admits a complete CAT(0) polyhedral metric is pseudo-collarable, its fundamental group at infinity is strongly perfectly semistable and has vanishing Chapman-Siebenmann…
In this project we explore the geometry of general metric spaces, where we do not necessarily have the tools of differential geometry on our side. Some metric spaces $(X,d)$ allow us to define geodesics, permitting us to compare geodesic…