Related papers: Uniqueness of maximum three-distance sets in the t…
Let $X$ be a general complex projective hypersurface in $\mathbb{P}^{n+1}$ of degree $d>1$. A point $P$ not in $X$ is called uniform if the monodromy group of the projection of $X$ from $P$ is isomorphic to the symmetric group. We prove…
The average distance of the equal hard spheres is introduced to evaluate the density of a given arrangement. The absolute smallest value is two radii because the spheres can not be closer to each other than their diameter. The absolute…
In the first part of this paper we study a best approximation of a vector in Euclidean space R^n with respect to a closed semi-algebraic set C and a given semi-algebraic norm. Assuming that the given norm and its dual norm are…
In this paper, we prove the following result : let X be a complex manifold, hyperbolic for the Carath\'eodory distance and let U be an open set relatively compact in X. Then, there exists k<1 such that we get, for the Carath\'eodory…
We recall the definition of an r-maximal set in a metric space as a maximal subset of diameter r. In the special case when the metric space is Euclidean such a set is exactly a solid of constant diameter r. In the process of reviewing the…
We provide conjectural necessary and (separately) sufficient conditions for the Hilbert scheme of points of a given length to have the maximum dimension tangent space at a point. The sufficient condition is claimed for 3D and reduces the…
In this paper we consider piecewise linear (pl) isometric embeddings of Euclidean polyhedra into Euclidean space. A Euclidean polyhedron is just a metric space $\mathcal{P}$ which admits a triangulation $\mathcal{T}$ such that each…
Motivated by integral point sets in the Euclidean plane, we consider integral point sets in affine planes over finite fields. An integral point set is a set of points in the affine plane $\mathbb{F}_q^2$ over a finite field $\mathbb{F}_q$,…
Huemer et al. (Discrete Mathematics, 2019) proved that for any two point sets $R$ and $B$ with $|R|=|B|$, the perfect matching that matches points of $R$ with points of $B$, and maximizes the total \emph{squared} Euclidean distance of the…
Recently, generalizations of the classical Three Gap Theorem to higher dimensions attracted a lot of attention. In particular, upper bounds for the number of nearest neighbor distances have been established for the Euclidean and the maximum…
Retaining the combinatorial Euclidean structure of a regular icosahedron, namely the 20 equiangular (planar) triangles, the 30 edges of length 1, and the 12 different vertices together with the incidence structure, we investigate variations…
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…
We prove that sets with positive upper Banach density in sufficiently large dimensions contain congruent copies of all sufficiently large dilates of three specific higher-dimensional patterns. These patterns are: $2^n$ vertices of a fixed…
An isometry is a geometric transformation that preserves distances between pairs of points. We present methods to classify isometries in the Euclidean plane, and extend these methods to spherical, single elliptical, and hyperbolic geometry.…
The famous Erd\H{o}s distinct distances problem asks the following: how many distinct distances must exist between a set of $n$ points in the plane? There are many generalisations of this question that ask one to consider different spaces…
For a given connected set $\Gamma$ in $d-$dimensional Euclidean space, we construct a connected set $\tilde\Gamma\supset \Gamma$ such that the two sets have comparable Hausdorff length, and the set $\tilde\Gamma$ has the property that it is…
In this article, we introduce a notion of curvature, denoted by $ k_X(T)$, for a metric triple $T$ inside a (possibly discrete) metric space $X$. Such a notion enables us to consider curvature information of any metric space, including…
We examine properties of equidistant sets determined by nonempty disjoint compact subsets of a compact 2-dimensional Alexandrov space (of curvature bounded below). The work here generalizes many of the known results for equidistant sets…
The three-dimensional linear regression problem is a problem of finding a spacial straight line best fitting a group of points in three-dimensional Euclidean space. This problem is considered in the present paper and a solution to it is…
The convex hull on three points in two dimensional euclidean space of three flat edges (trihedron) was studied. The Bohr-Sommerfeld quantization of the area of space is performed. It is shown that it reproduces exactly the equidistant…