Related papers: Finite Quasihypermetric Spaces

Let $(X, d)$ be a compact metric space and let $\mathcal{M}(X)$ denote the space of all finite signed Borel measures on $X$. Define $I \colon \mathcal{M}(X) \to \R$ by \[ I(\mu) = \int_X \int_X d(x,y) d\mu(x) d\mu(y), \] and set $M(X) =…

Let $(X, d)$ be a compact metric space and let $\mathcal{M}(X)$ denote the space of all finite signed Borel measures on $X$. Define $I \colon \mathcal{M}(X) \to \R$ by \[I(\mu) = \int_X \int_X d(x,y) d\mu(x) d\mu(y),\] and set $M(X) = \sup…

Let $(X, d)$ be a compact metric space and let $\mathcal{M}(X)$ denote the space of all finite signed Borel measures on $X$. Define $I \colon \mathcal{M}(X) \to \R$ by \[ I(\mu) = \int_X \int_X d(x,y) d\mu(x) d\mu(y), \] and set $M(X) =…

Let $Q$ denote the space of signed measures on the Borel $\sigma$-algebra of a separable complete space $X$. We endow $Q$ with the norm $\|q\|=\sup|\int\phi dq|$, where the supremum is taken over all Lipschitz with constant 1 functions…

The concept of a quasi-metric space arises by relaxing the requirement of the symmetry axiom in the definition of a metric. This small variation alters several structural properties possessed by a standard metric space. This article aims to…

Let $(X,d_X,\mu)$ be a metric measure space where $X$ is locally compact and separable and $\mu$ is a Borel regular measure such that $0 <\mu(B(x,r)) <\infty$ for every ball $B(x,r)$ with center $x \in X$ and radius $r>0$. We define…

For a metrizable space $X$ and a finite measure space $(\Omega,\mathfrak{M},\mu)$ let $M_{\mu}(X)$ and $M^f_{\mu}(X)$ be the spaces of all equivalence classes (under the relation of equality almost everywhere mod $\mu$) of…

We consider extensions of quasiconformal maps and the uniformization theorem to the setting of metric spaces $X$ homeomorphic to $\mathbb R^2$. Given a measure $\mu$ on such a space, we introduce $\mu$-quasiconformal maps $f:X \to \mathbb…

For a probability measure space $(X,\mathscr{A},\mu)$, we define a pseudometric $\delta$ on the ring $\mathcal{M}(X,\mathscr{A})$ of real-valued measurable functions on $X$ as $\delta(f,g)=\mu(X\setminus Z(f-g))$ and denote the topological…

Our main result states that, given a finite-dimensional vector space $E$, the pseudometric defined in the set of continuous quasinorms $\mathcal{Q}_0=\{\|\cdot\|:E\to\mathbb{R}\}$ as $$d(\|\cdot\|_X,\|\cdot\|_Y)=\min\{\mu:\|\cdot\|_X…

If $(X,d)$ is a Polish metric space of dimension $0$, then by Wadge's lemma, no more than two Borel subsets of $X$ can be incomparable with respect to continuous reducibility. In contrast, our main result shows that for any metric space…

We say that a finite metric space $X$ can be embedded almost isometrically into a class of metric spaces $C$, if for every $\epsilon > 0$ there exists an embedding of $X$ into one of the elements of $C$ with the bi-Lipschitz distortion less…

It is shown that if a compact metric space $(X, d)$ is bi-H\"older equivalent to an ultrametric space, then the logarithmic ratio $R(X,d)$ is finite. Conversely, if the logarithmic ratio $R(X,d)$ is finite and ${\A}^*_p (X) \ne \emptyset$…

In this paper, using the existence of infinite equidistant subsets of closed balls, we characterize the injectivity of ultrametric spaces for finite ultrametric spaces, which also gives a characterization of the Urysohn universal…

We study the set M(X) of full non-atomic Borel (finite or infinite) measures on a non-compact locally compact Cantor set X. For an infinite measure $\mu$ in M(X), the set $\mathfrak{M}_\mu = \{x \in X : {for any compact open set} U \ni x…

A metric space $(M, d)$ is said to be universal for a class of metric spaces if all metric spaces in the class can be isometrically embedded into $(M, d)$. In this paper, for a metrizable space $Z$ possessing abundant subspaces, we first…

Let $(X, \mathscr{L}, \lambda)$ and $(Y, \mathscr{M}, \mu)$ be finite measure spaces for which there exist $A \in \mathscr{L}$ and $B \in \mathscr{M}$ with either $0 < \lambda(A) < 1 < \lambda(X)$ and $0 < \mu(B) < \mu(Y)$, or the other way…

A metric space $\mathrm{M}=(M,\de)$ is {\em indivisible} if for every colouring $\chi: M\to 2$ there exists $i\in 2$ and a copy $\mathrm{N}=(N, \de)$ of $\mathrm{M}$ in $\mathrm{M}$ so that $\chi(x)=i$ for all $x\in N$. The metric space…

Let $P$ be a directed set and $X$ a space. A collection $\mathcal{C}$ of subsets of $X$ is \emph{$P$-locally finite} if $\mathcal{C}=\bigcup \{ \mathcal{C}_p : p \in P\}$ where (i) if $p \le p'$ then $\mathcal{C}_p \subseteq…

Let $(X, d)$ be an ultrametric space and let $d_H$ be the Hausdorff distance on the set $\bar{\mathbf{B}}_X$ of all closed balls in $(X, d)$. Some interconnections between the properties of the spaces $(X, d)$ and $(\bar{\mathbf{B}}_X,…