Related papers: Two nearly equal distances in $R^d$

Let $\{p_1, \ldots , p_n \} \subset {\Bbb{R}}^2$ be a separated point set, i.e., any two points have a distance at least $1$. Let $k \ge 1$ be an integer, and $1 \le t_1 < \ldots < t_k$ be real numbers. Let $\delta > 0$. Suppose for all $1…

A set in $\mathbb R^d$ is called almost-equidistant if for any three distinct points in the set, some two are at unit distance apart. First, we give a short proof of the result of Bezdek and L\'angi claiming that an almost-equidistant set…

We say that a set of points $S\subset \mathbb{R}^d$ is an $\varepsilon$-nearly $k$-distance set if there exist $1\le t_1\le \ldots\le t_k,$ such that the distance between any two distinct points in $S$ falls into…

The Erd\H os unit distance conjecture in the plane says that the number of pairs of points from a point set of size $n$ separated by a fixed (Euclidean) distance is $\leq C_{\epsilon} n^{1+\epsilon}$ for any $\epsilon>0$. The best known…

Let $S$ be a set of points in $\mathbb{R}^2$ contained in a circle and $P$ an unrestricted point set in $\mathbb{R}^2$. We prove the number of distinct distances between points in $S$ and points in $P$ is at least…

In this paper we determine the maximum number of points in $\mathbb{R}^d$ which form exactly $t$ distinct triangles, where we restrict ourselves to the case of $t = 1$. We denote this quantity by $F_d(t)$. It was known from the work of…

Let $X$ be an $n$--element finite set, $0<k\leq n/2$ an integer. Suppose that $\{A_1,A_2\} $ and $\{B_1,B_2\} $ are pairs of disjoint $k$-element subsets of $X$ (that is, $|A_1|=|A_2|=|B_1|=|B_2|=k$, $A_1\cap A_2=\emptyset$, $B_1\cap…

Since ancient times mathematicians consider geometrical objects with integral side lengths. We consider plane integral point sets $\mathcal{P}$, which are sets of $n$ points in the plane with pairwise integral distances where not all the…

We consider the number of distinct distances between two finite sets of points in ${\bf R}^k$, for any constant dimension $k\ge 2$, where one set $P_1$ consists of $n$ points on a line $l$, and the other set $P_2$ consists of $m$ arbitrary…

Let $P$ be a set of $n$ points in $\mathbb{R}^d$, in general position. We remove all of them one by one, in each step erasing one vertex of the convex hull of the current remaining set. Let $g_d(P)$ denote the number of different removal…

For a positive integer $d$, a set of points in $d$-dimensional Euclidean space is called almost-equidistant if for any three points from the set, some two are at unit distance. Let $f(d)$ denote the largest size of an almost-equidistant set…

We show that the maximum number of unit distances or of diameters in a set of n points in d-dimensional Euclidean space is attained only by specific types of Lenz constructions, for all d >= 4 and n sufficiently large, depending on d. As a…

We show that there are sets of $n$ points in the plane with $n$ arbitrarily large that contain more than $n^{1.014}$ pairs of points separated by a distance exactly $1$. This improves on very recent work of a team at OpenAI, who proved the…

A finite set of points in $\mathbb R^d$ is called almost-equidistant if among any three distinct points in the set, some two are at unit distance. We prove that an almost-equidistant set in $\mathbb R^d$ has cardinality at most $5d^{13/9}$.

Let $d_1\leq d_2\leq\ldots\leq d_{n\choose 2}$ denote the distances determined by $n$ points in the plane. It is shown that $\min\sum_i (d_{i+1}-d_i)^2=O(n^{-6/7})$, where the minimum is taken over all point sets with minimal distance $d_1…

A homogeneous set of $n$ points in the $d$-dimensional Euclidean space determines at least $\Omega(n^{2d/(d^2+1)} / \log^{c(d)} n)$ distinct distances for a constant $c(d)>0$. In three-space, we slightly improve our general bound and show…

Given a real number $t>1$, a geometric $t$-spanner is a geometric graph for a point set in $\mathbb{R}^d$ with straight lines between vertices such that the ratio of the shortest-path distance between every pair of vertices in the graph…

A double-normal pair of a finite set $S$ of points from $R^d$ is a pair of points $\{p,q\}$ from $S$ such that $S$ lies in the closed strip bounded by the hyperplanes through $p$ and $q$ perpendicular to $pq$. A double-normal pair $pq$ is…

Consider a dataset of n(d) points generated independently from R^d according to a common p.d.f. f_d with support(f_d) = [0,1]^d and sup{f_d([0,1]^d)} growing sub-exponentially in d. We prove that: (i) if n(d) grows sub-exponentially in d,…

For the unit sphere S^d in Euclidean space R^(d+1), we show that for d-1<s<d and any N>1, discrete N-point minimal Riesz s-energy configurations are well separated in the sense that the minimal distance between any pair of distinct points…