Related papers: On computable classes of equidistant sets: multiva…
This short note gathers known results to state that the squared distance function to a (nonconvex) closed set of an Euclidean space is Polyak-{\L}ojasiewicz. As a fuzzy reciprocate, we also recall that every Polyak-{\L}ojasiewicz function…
Functions with uniform sublevel sets can represent orders, preference relations or other binary relations and thus turn out to be a tool for scalarization that can be used in multicriteria optimization, decision theory, mathematical…
A subset $S$ of vertices, in a connected graph $G$, is called a distance-equalizer set if for every pair of distinct vertices outside $S$, there exists a vertex in $S$ equidistant to both. The equidistant dimension, denoted by $\xi(G)$, is…
Evenly convex sets in a topological vector space are defined as the intersection of a family of open half spaces. We introduce a generalization of this concept in the conditional framework and provide a generalized version of the bipolar…
Defining distances over finite fields formally by $||x-y||:=(x_1-y_1)^2+\cdots + (x_d-y_d)^2$ for $x,y\in \mathbb{F}_q^d$, distance problems naturally arise in analogy to those studied by Erd\H{o}s and Falconer in Euclidean space. Given a…
In this article a relation between curvature functionals for surfaces in the Euclidean space and area functionals in relative differential geometry will be given. Relative differential geometry can be described as the geometry of surfaces…
A theorem is proved concerning approximation of analytic functions by multivariate polynomials in the $s$-dimensional hypercube. The geometric convergence rate is determined not by the usual notion of degree of a multivariate polynomial,…
We analyse the geometric properties of the high derivatives of the distance function from a submanifold of the Euclidean space. In particular, we show some relations with the second fundamental form and its covariant derivatives of…
Given two arbitrary closed sets in Euclidean space, a simple transversality condition guarantees that the method of alternating projections converges locally, at linear rate, to a point in the intersection. Exact projection onto nonconvex…
Let M be a smooth strictly convex closed surface in space and denote by H the set of points x in the exterior of M such that all the tangent segments from x to M have equal lengths. In this note we prove that if H is either a closed surface…
We introduce a novel type of approximation spaces for functions with values in a nonlinear manifold. The discrete functions are constructed by piecewise polynomial interpolation in a Euclidean embedding space, and then projecting pointwise…
In this article, we introduce a finite element method designed for the robust computation of approximate signed distance functions to arbitrary boundaries in two and three dimensions. Our method employs a novel prediction-correction…
The purpose of this paper is to give a survey on the notions of distance between subsets either of a metric space or of a measure space, including definitions, a classification, and a discussion of the best-known distance functions, which…
The equichordal point problem is a classical question in geometry, asking whether there exist multiple equichordal points within a single convex body. An equichordal point is defined as a point through which all chords of the convex body…
Convex geometry has recently attracted great attention as a framework to formulate general probabilistic theories. In this framework, convex sets and affine maps represent the state spaces of physical systems and the possible dynamics,…
We study the computation of local approximations of invariant manifolds of parabolic fixed points and parabolic periodic orbits of periodic vector fields. If the dimension of these manifolds is two or greater, in general, it is not possible…
In the $m$-dimensional affine space $AG(m,q)$ over the finite field $\mathbb{F}_q$ of odd order $q$, the analogous of the Euclidean distance gives rise to a graph $\mathfrak{G}_{m,q}$ where vertices are the points of $AG(m,q)$ and two…
The point-to-set principle of J. Lutz and N. Lutz (2018) has recently enabled the theory of computing to be used to answer open questions about fractal geometry in Euclidean spaces $\mathbb{R}^n$. These are classical questions, meaning that…
The present work concerns generalized convex sets in the real multi-dimensional Euclidean space, known as weakly $1$-convex and weakly $1$-semiconvex sets. An open set is called weakly $1$-convex (weakly $1$-semiconvex) if, through every…
One of the oldest and richest problems from continuous location science is the famous Fermat-Torricelli problem, asking for the unique point in Euclidean space that has minimal distance sum to n given (non-collinear) points. Many natural…