Related papers: Set-valued metrics and generalized Hausdorff dista…

We study the distance on the Bruhat-Tits building of the group $\mathrm{SL}_d(\mathbb{Q}_p)$ (and its other combinatorial properties). Coding its vertices by certain matrix representatives, we introduce a way how to build formulas with…

This paper presents methods to compare high order networks, defined as weighted complete hypergraphs collecting relationship functions between elements of tuples. They can be considered as generalizations of conventional networks where only…

We develop a new class of distances for objects including lines, hyperplanes, and trajectories, based on the distance to a set of landmarks. These distances easily and interpretably map objects to a Euclidean space, are simple to compute,…

We study closed sets $F \subset {\mathbb R}^d$ whose distance function $d_F:= {\rm dist}\,(\cdot,F)$ is DC (i.e., is the difference of two convex functions on ${\mathbb R}^d$). Our main result asserts that if $F \subset {\mathbb R}^2$ is a…

The class of convex sets that admit approximations as Minkowski sum of a compact convex set and a closed convex cone in the Hausdorff distance is introduced. These sets are called approximately Motzkin-decomposable and generalize the notion…

We prove that if $A$ is a Borel set in the plane of equal Hausdorff and packing dimension $s>1$, then the set of pinned distances $\{ |x-y|:y\in A\}$ has full Hausdorff dimension for all $x$ outside of a set of Hausdorff dimension $1$ (in…

It is shown that any bounded metric space can be isometrically embedded into the Gromov--Hausdorff metric class GH. This result is a consequence of local geometry description of the class GH in a sufficiently small neighborhood of a generic…

We prove new bounds on the dimensions of distance sets and pinned distance sets of planar sets. Among other results, we show that if $A\subset\mathbb{R}^2$ is a Borel set of Hausdorff dimension $s>1$, then its distance set has Hausdorff…

Let $(X,d,\mu)$ be a space of homogeneous type. In this note we study the relationship between two types of $s$-sets: relative to a distance and relative to a measure. We find a condition on a closed subset $F$ of $X$ under which we have…

The present paper generalizes the result from one of the papers by Galstyan. Namely, we consider two nonempty subsets $A$ and $B$ of a metric space $X$, and construct one-parametric family $F_r$ of subsets obtained by intersection between…

We characterize the surjective isometries, with respect to the Hausdorff distance, of the class of bodies given by intersections of Euclidean unit balls. We show that any such isometry is given by the composition of a rigid motion with…

We give a complete description of the possible Hausdorff dimensions of escaping sets for meromorphic functions with a finite number of singular values. More precisely, for any given $d\in [0,2]$ we show that there exists such a meromorphic…

The author has recently introduced the class of CNED sets in Euclidean space, generalizing the classical notion of NED sets, and shown that they are quasiconformally removable. A set $E$ is CNED if the conformal modulus of a curve family is…

Distance-based metrics, such as the Hausdorff distance (HD), are widely used to validate segmentation performance in (bio)medical imaging. However, their implementation is complex, and critical differences across open-source tools remain…

We discuss a method to estimate the measure of a compact set which is approximated using the Hausdorff distance by a sequence of compact sets. We do this by considering corresponding fattenings of the sequence of compact sets and showing…

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) =…

We introduce a class of Falconer distance problems, which we call of restricted type, lying between the classical version and its pinned variant. Prototypical restricted distance sets are the diagonal distance sets, $k$-point configuration…

In this short note, we show that the distance function to any finite set $X\subset \mathbb{R}^n$ is a topological Morse function, regardless of whether $X$ is in general position. We also precisely characterize its topological critical…

For each given $p\in[1,\infty]$ we investigate certain sub-family $\mathcal{M}_p$ of the collection of all compact metric spaces $\mathcal{M}$ which are characterized by the satisfaction of a strengthened form of the triangle inequality…

Two objects may be close in the Hausdorff metric, yet have very different geometric and topological properties. We examine other methods of comparing digital images such that objects close in each of these measures have some similar…