Related papers: Nearly equal distances in the plane, II

In this paper, we derive formulas for the analytical calculation of the moments of the distance between two uniformly and independently distributed random points in an $n$-sided regular polygon. A number of closed form expressions is…

Using elementary methods, we prove new formulas for $\operatorname{pp}(n)$, the number of plane partitions of $n$, $\operatorname{pp}_r(n)$, the number of plane partitions of $n$ with at most $r$ rows, $\operatorname{pp}^s(n)$, the number…

In general, it is difficult to measure distances in the Weil-Petersson metric on Teichm\"uller space. Here we consider the distance between strata in the Weil-Petersson completion of Teichm\"uller space of a surface of finite type. Wolpert…

In a countably normed space which is a linear space equipped with a countable number of pair-wise compatible norms, we prove the existence of a common nearest point (in all norms) from a point outside a nonempty subset if this subset is…

If two closed Jordan curves in the plane have precisely one point in common, then it is called a {\em touching point}. All other intersection points are called {\em crossing points}. The main result of this paper is a Crossing Lemma for…

Let $B$ and $R$ be point sets (of {\em blue} and {\em red} points, respectively) in the plane, such that $P:=B\cup R$ is in general position, and $|P|$ is even. A line $\ell$ is {\em balanced} if it spans one blue and one red point, and on…

We prove that for all positive integers $n$ and $k$, there exists an integer $N = N(n,k)$ satisfying the following. If $U$ is a set of $k$ direction vectors in the plane and $\mathcal{J}_U$ is the set of all line segments in direction $u$…

Given $n$ continuous open curves in the plane, we say that a pair is touching if they have only one interior point in common and at this point the first curve does not get from one side of the second curve to its other side. Otherwise, if…

We study the distances of edges within cliques in a soft random geometric graph on a torus, where the vertices are points of a homogeneous Poisson point process, and far-away points are less likely to be connected than nearby points. We…

Applications in machine learning and data mining require computing pairwise Lp distances in a data matrix A. For massive high-dimensional data, computing all pairwise distances of A can be infeasible. In fact, even storing A or all pairwise…

Consider a walk in the plane made of $n$ unit steps, with directions chosen independently and uniformly at random at each step. Rayleigh's theorem asserts that the probability for such a walk to end at a distance less than 1 from its…

The Ulam distance of two permutations on $[n]$ is $n$ minus the length of their longest common subsequence. In this paper, we show that for every $\varepsilon>0$, there exists some $\alpha>0$, and an infinite set $\Gamma\subseteq…

Given a set $P$ of $n$ points in the plane and a multiset $W$ of $k$ weights with $k\leq n$, we assign each weight in $W$ to a distinct point in $P$ to minimize the maximum weighted distance from the weighted center of $P$ to any point in…

Let $G$ be a complete $k$-partite simple undirected graph with parts of sizes $p_1\le p_2...\le p_k$. Let $P_j=\sum_{i=1}^jp_i$ for $j=1,...,k$. It is conjectured that $G$ has distance magic labeling if and only if $\sum_{i=1}^{P_j}…

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…

The {\em bottleneck distance} is a natural measure of the distance between two finite point sets of equal cardinality, defined as the minimum over all bijections between the point sets of the maximum distance between any pair of points put…

We consider the problem of computing the (two-sided) Hausdorff distance between the unit $\ell_{p_{1}}$ and $\ell_{p_{2}}$ norm balls in finite dimensional Euclidean space for $1 \leq p_1 < p_2 \leq \infty$, and derive a closed-form formula…

For a convex body $K\subset\mathbb{R}^d$ the mean distance $\Delta(K)=\mathbb{E}|X_1-X_2|$ is the expected Euclidean distance of two independent and uniformly distributed random points $X_1,X_2\in K$. Optimal lower and upper bounds for…

Given a positive real value $\delta$, a set $P$ of points along a line and a distance function $d$, in the movement to independence problem, we wish to move the points to new positions on the line such that for every two points $p_{i},p_{j}…

Let $x_1, \dots, x_n$ be points in a metric space and define the distance matrix $D \in \mathbb{R}^{n \times n}$ by ${D}_{ij} = d(x_i, x_j)$. The Perron-Frobenius Theorem implies that there is an eigenvector $v \in \mathbb{R}^n_{}$ with…