Let $\sigma _q \,:\,{{\mathbb{R}}^q} \to{\textbf{S}}^q\setminus N_q$ be the inverse of the stereographic projection with center the north pole $N_q$. Let $W_i$ be a closed subset of ${\mathbb{R}}^{q_i}$, for $i=1,2$. Let $\Phi \,:\,W_1 \to W_2$ be a bi-Lipschitz homeomorphism. The main result states that the homeomorphism $\sigma _{q_2}\circ \Phi \circ \sigma _{q_1}^{-1}$ is a bi-Lipschitz homeomorphism, extending bi-Lipschitz-ly at $N_{q_1}$ with value $N_{q_2}$ whenever $W_1$ is unbounded.

As two straightforward applications in the polynomially bounded o-minimal context over the real numbers, we obtain for free a version at infinity of: (1) Sampaio’s tangent cone result and (2) links preserving re-parametrization of definable bi-Lipschitz homeomorphisms of Valette.

Magnitude is a numerical invariant of compact metric spaces, originally inspired by category theory and now known to be related to myriad other geometric quantities. Generalizing earlier results in $\ell _1^n$ and Euclidean space, we prove an upper bound for the magnitude of a convex body in a hypermetric normed space in terms of its Holmes–Thompson intrinsic volumes. As applications of this bound, we give short new proofs of Mahler’s conjecture in the case of a zonoid and Sudakov’s minoration inequality.

Let $(G,+)$ be a compact, abelian, and metrizable topological group. In this group we take $g\in G$ such that the corresponding automorphism $\tau _g$ is ergodic. The main result of this paper is a new ergodic theorem for functions in $L^1(G,M)$ , where M is a Hadamard space. The novelty of our result is that we use inductive means to average the elements of the orbit $\{\tau _g^n(h)\}_{n\in \mathbb {N}}$ . The advantage of inductive means is that they can be explicitly computed in many important examples. The proof of the ergodic theorem is done firstly for continuous functions, and then it is extended to $L^1$ functions. The extension is based on a new construction of mollifiers in Hadamard spaces. This construction has the advantage that it only uses the metric structure and the existence of barycenters, and does not require the existence of an underlying vector space. For this reason, it can be used in any Hadamard space, in contrast to those results that need to use the tangent space or some chart to define the mollifier.

We introduce properties of metric spaces and, specifically, finitely generated groups with word metrics, which we call coarse coherence and coarse regular coherence. They are geometric counterparts of the classical algebraic notion of coherence and the regular coherence property of groups defined and studied by Waldhausen. The new properties can be defined in the general context of coarse metric geometry and are coarse invariants. In particular, they are quasi-isometry invariants of spaces and groups. The new framework allows us to prove structural results by developing permanence properties, including the particularly important fibering permanence property, for coarse regular coherence.

Let $\mathbf{TB}$ be the category of totally bounded, locally compact metric spaces with the ${{C}_{0}}$ coarse structures. We show that if $X$ and $Y$ are in $\mathbf{TB}$, then $X$ and $Y$ are coarsely equivalent if and only if their Higson coronas are homeomorphic. In fact, the Higson corona functor gives an equivalence of categories $\mathbf{TB}\,\to \,\mathbf{K}$, where $\mathbf{K}$ is the category of compact metrizable spaces. We use this fact to show that the continuously controlled coarse structure on a locally compact space $X$ induced by some metrizable compactification $\widetilde{X}$ is determined only by the topology of the remainder $\widetilde{X}\,\backslash \,X$.

A metric space $\text{M=}\left( M;\text{d} \right)$ is homogeneous if for every isometry $f$ of a finite subspace of $\text{M}$ to a subspace of $\text{M}$ there exists an isometry of $\text{M}$ onto $\text{M}$ extending $f$. The space $\text{M}$ is universal if it isometrically embeds every finite metric space $\text{F}$ with $\text{dist}\left( \text{F} \right)\subseteq \text{dist}\left( \text{M} \right)$ (with $\text{dist}\left( \text{M} \right)$ being the set of distances between points in $\text{M}$).

A metric space $U$ is a Urysohn metric space if it is homogeneous, universal, separable, and complete. (We deduce as a corollary that a Urysohn metric space $U$ isometrically embeds every separable metric space $\text{M}$ with $\text{dist}\left( \text{M} \right)\subseteq \text{dist}\left( U \right)$.)

The main results are: (1) A characterization of the sets $\text{dist}\left( U \right)$ for Urysohn metric spaces $U$. (2) If $R$ is the distance set of a Urysohn metric space and $\text{M}$ and $\text{N}$ are two metric spaces, of any cardinality with distances in $R$, then they amalgamate disjointly to a metric space with distances in $R$. (3) The completion of every homogeneous, universal, separable metric space $\text{M}$ is homogeneous.

By $\text{d(X,Y)}$ we denote the (multiplicative) Banach–Mazur distance between two normed spaces $X$ and $Y$. Let $X$ be an $n$-dimensional normed space with $\text{d(X,}\,l_{\infty }^{n}\text{)}\,\le \,\text{2}$, where $l_{\infty }^{n}$ stands for ${{\mathbb{R}}^{n}}$ endowed with the norm $\parallel ({{x}_{1}},\,.\,.\,.\,,\,{{x}_{n}}){{\parallel }_{\infty }}\,:=\,\max \{|{{x}_{1}}|,\,.\,.\,.\,,\,|{{x}_{n}}|\}$. Then every metric space $(S,\,\rho )$ of cardinality $n+1$ with norm $\rho $ satisfying the condition $\max D/\min D\,\le \,2/\,\text{d(}X,\,l_{\infty }^{n}\text{)}$ for $D\,:=\,\{\rho (a,\,b)\,:\,a,\,b\,\in \,S,\,a\,\ne \,b\}$ can be isometrically embedded into $X$.

We consider Ptolemy's inequality in a metric space setting. It is not hard to see that CAT(0) spaces satisfy this inequality. Although the converse is not true in full generality, we show that if our Ptolemaic space is either a Riemannian or Finsler manifold, then it must also be CAT(0). Ptolemy's inequality is closely related to inversions of metric spaces. We exploit this link to establish a new characterization of Euclidean space amongst all Riemannian manifolds.

A projective metric space is a pappian projective space together with a quadric and a certain equivalence relation on the pairs of those points which do not belong to the quadric. This equivalence relation is defined by means of the corresponding quadratic form and satisfies a condition which is a projective version of Miquel's theorem. We characterize the projective metric spaces of dimension at least two over fields of order at least 13.