Related papers: Bounds for the minimum diameter of integral point …
We prove a bound for the geodesic diameter of a subset of the unit ball in $\mathbb{R}^n$ described by a fixed number of quadratic equations and inequalities, which is polynomial in $n$, whereas the known bound for general degree is…
In this paper, we study the problem of computing a minimum-width double-strip or parallelogram annulus that encloses a given set of $n$ points in the plane. A double-strip is a closed region in the plane whose boundary consists of four…
In this work, we carry out structural and algorithmic studies of a problem of barrier forming: selecting theminimum number of straight line segments (barriers) that separate several sets of mutually disjoint objects in the plane. The…
For $\varepsilon\in(0,1/2)$ and a natural number $d\ge 2$, let $N$ be a natural number with \[ N \,\ge\, 2^9\,\log_2(d)\, \left(\frac{\log_2(1/\varepsilon)}{\varepsilon}\right)^2. \] We prove that there is a set of $N$ points in the unit…
Families of boxes in $\mathbb R^d$ are considered. In the paper an upper bound on the size of a minimum transversal in terms of the space dimension and the independence number of the given family was improved.
We study the problem of minimum enclosing rectangle with outliers, which asks to find, for a given set of $n$ planar points, a rectangle with minimum area that encloses at least $(n-t)$ points. The uncovered points are regarded as outliers.…
We use computational experiments to find the rectangles of minimum area into which a given number n of non-overlapping congruent circles can be packed. No assumption is made on the shape of the rectangles. Most of the packings found have…
We develop a Morse-Lusternik-Schnirelmann theory for the distance between two points of a smoothly embedded circle in a complete Riemannian manifold. This theory suggests very naturally a definition of width that generalises the classical…
We study the asymptotic behavior of the maximum interpoint distance of random points in a $d$-dimensional set with a unique diameter and a smooth boundary at the poles. Instead of investigating only a fixed number of $n$ points as $n$ tends…
Given a non-empty bounded subset of hyperbolic space and a Kleinian group acting on that space, the orbital set is the orbit of the given set under the action of the group. We may view orbital sets as bounded (often fractal) subsets of…
The inaccessibility of a point p in a bounded domain D \subset R^n is the minimum of the lengths of segments through p with boundary at \bd D. The points of maximum inaccessibility I_D are those where the inaccessibility achieves its…
The injectivity radius of a manifold is an important quantity, both from a theoretical point of view and in terms of numerical applications. It is the largest possible radius within which all geodesics are unique and length-minimizing. In…
Let X be a group with identity e, let A be an infinite set of generators for X, and let (X,d_A) be the metric space with the word metric d_A induced by A. If the diameter of the space is infinite, then for every positive integer h there are…
Integral circulant graphs are proposed as models for quantum spin networks that permit a quantum phenomenon called perfect state transfer. Specifically, it is important to know how far information can potentially be transferred between…
We consider the situation where one is given a set S of points in the plane and a collection D of unit disks embedded in the plane. We show that finding a minimum cardinality subset of D such that any path between any two points in S is…
A set $X$ in the Euclidean space $\mathbb{R}^d$ is called an $m$-distance set if the set of Euclidean distances between two distinct points in $X$ has size $m$. An $m$-distance set $X$ in $\mathbb{R}^d$ is said to be maximal if there does…
The Erd\H{o}s distinct distance problem is a ubiquitous problem in discrete geometry. Somewhat less well known is Erd\H{o}s' distinct angle problem, the problem of finding the minimum number of distinct angles between $n$ non-collinear…
In this article, we consider the problems of finding in $d+1$ dimensions a minimum-volume axis-parallel box, a minimum-volume arbitrarily-oriented box and a minimum-volume convex body into which a given set of $d$-dimensional unit-radius…
For a $d$-dimensional polytope with $v$ vertices, $d+1\le v\le2d$, we calculate precisely the minimum possible number of $m$-dimensional faces, when $m=1$ or $m\ge0.62d$. This confirms a conjecture of Gr\"unbaum, for these values of $m$.…
A point set $P \subset {\Bbb{R}}^d$ is {\it separated} if the minimum distance between any two points in $P$ is at least $1$. For $d \ne 4,5,$ we determine, for every $t_1,t_2 \ge 1$, and for $n$ at least a suitable $n_d$, the maximum…