Related papers: A note on ANR's

We prove explicit bounds on the radius of a ball centered at the origin which is guaranteed to contain all bounded connected components of a semi-algebraic set $S \subset \mathbbm{R}^k$ defined by a quantifier-free formula involving $s$…

A Randers space is a differentiable manifold equipped with a Randers metric. It is the sum of a Riemannian metric and a one-form on the base manifold. The compatibility of a linear connection with the metric means that the parallel…

We describe the proper absolute (neighborhood) extensors for the class of at most $n$-dimensional spaces, notation $\rm{A(N)E}_p(n)$. For example, the unique locally compact $n$-dimensional separable metric space $X\in\rm{AE}_p(n)$…

A $graph$ $metric$ on a set $X$ is any function $d: E_d \to\mathbb R_+:=\{x\in\mathbb R:x>0\}$ defined on a connected graph $ E_d \subseteq[X]^2:=\{A\subseteq X:|A|=2\}$ and such that for every $\{x,y\}\in E_d$ we have $d(\{x,y\})\le\hat…

Let A be the annulus in R^2 centered at the origin with inner and outer radii r(1-\epsilon) and r, respectively. Place points {x_i} in R^2 according to a Poisson process with intensity 1 and let G_A be the random graph with vertex set {x_i}…

For each vector $x\in \ell^{\infty}$, we can define the non-empty compact set $L_x$ of accumulation points of $x$. Given an infinite subset $A$ of $\mathbb{N}\backslash\{1\}$, we can therefore investigate under which conditions on $A$, the…

For a metrizable space $X$, we denote by $\mathrm{Met}(X)$ the space of all metric that generate the same topology of $X$. The space $\mathrm{Met}(X)$ is equipped with the supremum distance. In this paper, for every strongly…

It is natural to ask for a reasonable constant k having the property that any open set of area greater than k on a bidimensional sphere of area 1 always contains the vertices of a regular tetrahedron. We shall prove that it is sufficient to…

We show that for any subset $A\subset [0,\infty)$, where $0\in A$, there exists a Bernstein set $X\subset \mathbb R$ such that $A$ is the center of distances of $X$.

In this note, we present examples of non-quasi-geodesic metric spaces which are hyperbolic (i.e., satisfying the Gromov's $4$-point condition) while the intersection of any two metric balls therein does not either "look like" a ball or has…

The paper is the last in the cycle devoted to the solution of Alexandrov's problem for non-positively curved spaces. Here we study non-positively curved spaces in the sense of Busemann. We prove that if $X$ is geodesically complete…

The sum theorem and its corollaries are proved for a countable family of zero-dimensional (in the sense of small and large inductive bidimensions) p-closed sets, using a new notion of relative normality whose topological correspondent is…

We investigate properties which remain invariant under the action of quasi-M\"obius maps of quasi-metric spaces. A metric space is called doubling with constant D if every ball of finite radius can be covered by at most D balls of half the…

We call a nonempty subset $A$ of a topological space $X$ finitely non-Urysohn if for every nonempty finite subset $F$ of $A$ and every family $\{U_x:x\in F\}$ of open neighborhoods $U_x$ of $x\in F$, $\cap\{\mathrm{cl}(U_x):x\in…

Let $X$ be a non-singular algebraic curve of genus $g$. We prove that the Brill-Noether locus $\bns $ is non-empty if $d= nd' +d'' $ with $0< d'' <2n$, $1\le s\le g$, $d'\geq (s-1)(s+g)/s $, $n\leq d''+(n-k)g$, $(d'',k)\ne(n,n)$. These…

For $p\in (1,\infty)$ let $\mathscr{P}_p(\mathbb{R}^3)$ denote the metric space of all $p$-integrable Borel probability measures on $\mathbb{R}^3$, equipped with the Wasserstein $p$ metric $\mathsf{W}_p$. We prove that for every…

Boundary analysis is developed for a rich class of generally infinite weighted graphs with compact metric completions. These graph completions have totally disconnected boundaries. The classical notion of $\epsilon$-components and the…

For a (compact) subset $K$ of a metric space and $\varepsilon > 0$, the {\em covering number} $N(K , \varepsilon )$ is defined as the smallest number of balls of radius $\varepsilon$ whose union covers $K$. Knowledge of the {\em metric…

We prove that if $M$ is an infinite complete metric space then the set of strongly norm-attaining Lipschitz functions $\SA(M)$ contains a linear subspace isomorphic to $c_0$. This solves an open question posed by V. Kadets and O. Rold\'an.

Let $X$ be a smooth dendroid in the plane $\mathbb R^2$. We show that each endpoint of $X$ is arcwise accessible from $\mathbb R^2\setminus X$, and that the space of endpoints $E(X)$ has the property of a circle. In the event that $E(X)$ is…