Related papers: Local vs. global Lipschitz geometry

We survey recent results on the local and global integrability of a Lie algebroid, as well as the integrability of infinitesimal multiplicative geometric structures on it.

We study a new bi-Lipschitz invariant \lambda(M) of a metric space M; its finiteness means that Lipschitz functions on an arbitrary subset of M can be linearly extended to functions on M whose Lipschitz constants are enlarged by a factor…

Heinrich Tietze has shown that for a closed connected subset of euclidean space being convex is a local property. We generalize this to CAT(0)-spaces and locally compact CAT(\kappa) spaces. As an application we give a construction of…

We study Finsler PL spaces, that is simplicial complexes glued out of simplices cut off from some normed spaces. We are interested in the class of Finsler PL spaces featuring local uniqueness of geodesics (for complexes made of Euclidean…

We propose the notions of uniform local weak o-minimality and $*$-local weak o-minimality. Local monotonicity theorems hold in definably complete locally o-minimal structures and uniformly locally o-minimal structures of the second kind. In…

Consider a definably complete uniformly locally o-minimal expansion of the second kind of a densely linearly ordered abelian group. Let $f:X \rightarrow R^n$ be a definable map, where $X$ is a definable set and $R$ is the universe of the…

We define a discrete closure operation for definably complete locally o-minimal structures $\mathcal M$. The pair of the underlying set of $\mathcal M$ and the discrete closure operation forms a pregeometry. We define the rank of a…

We prove that every open subset of a euclidean building is a finite dimensional absolute neighborhood retract. This implies in particular that such a set has the homotopy type of a finite dimensional simplicial complex. We also include a…

Let $M$ be a separable metric space. We say that $f=(f_n):M\to c_0$ is a good-$\lambda$-embedding if, whenever $x,y\in M$, $x\ne y$ implies $d(x,y)\le\Vert f(x)-f(y)\Vert$ and, for each $n$, $Lip(f_n)<\lambda$, where $Lip(f_n)$ denotes the…

For non-empty sets X we define notions of distance and pseudo metric with values in a partially ordered set that has a smallest element $\theta $. If $h_X$ is a distance in $X$ (respectively, a pseudo metric in $X$), then the pair $(X,h_X)$…

In this short note, we show that, in any given metric space, every Lipschitz open-map image of every subset of a given metric space whose boundary is Hausdorff-null is Hausdorff-measurable with respect to the same dimension. The main…

We characterize the notion of definable compactness for topological spaces definable in o-minimal structures, answering questions of Peterzil and Steinhorn (1999) and Johnson (2018). Specifically, we prove the equivalence of various…

We show that separability and second-countability are first-order properties among topological spaces definable in o-minimal expansions of $(\mathbb{R},<)$. We do so by introducing first-order characterizations -- definable separability and…

We introduce the notion of (almost isometric) local retracts in metric space as a natural non-linear version of the concepts of locally complemented and almost isometric ideals from Banach spaces. We prove that given two metric spaces…

We proof here the existence of a topological thick and thin decomposition of any closed definable thick isolated singularity germ in the spirit of the recently discovered metric thick and thin decomposition of complex normal surface…

We investigate a class of metrics for 2-manifolds in which, except for a discrete set of singular points, the metric is locally isometric to an L_1 (or equivalently L_infinity) metric, and show that with certain additional conditions such…

We show that every finite-dimensional Euclidean space contains compact universal differentiability sets of upper Minkowski dimension one. In other words, there are compact sets $S$ of upper Minkowski dimension one such that every Lipschitz…

Sharply o-minimal structures (denoted \so-minimal) are a strict subclass of the o-minimal structures, aimed at capturing some finer features of structures arising from algebraic geometry and Hodge theory. Sharp o-minimality associates to…

A vertex $v$ is said to distinguish two other vertices $x$ and $y$ of a nontrivial connected graph G if the distance from $v$ to $x$ is different from the distance from $v$ to $y$. A set $S\subseteq V(G)$ is a local metric set for $G$ if…

For a compact metric space $(K, \rho)$, the predual of $Lip(K, \rho)$ can be identified with the normed space $M(K)$ of finite (signed) Borel measures on $K$ equipped with the Kantorovich-Rubinstein norm, this is due to Kantorovich [20].…