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We prove that, for any covering of a unit d-dimensional Euclidean ball by smaller balls, the sum of radii of the balls from the covering is greater than d. We also investigate the problem of finding lower and upper bounds for the sum of powers of radii of the balls covering a unit ball. [ABSTRACT FROM AUTHOR]

We show that a jammed packing of disks with generic radii, in a generic container, is such that the minimal number of contacts occurs and there is only one dimension of equilibrium stresses, which have been observed with numerical Monte Carlo simulations. We also point out some connections to packings with different radii and results in the theory of circle packings whose graph forms a triangulation of a given topological surface. [ABSTRACT FROM AUTHOR]

We investigate the L 1 geodesic farthest neighbors in a simple polygon P, and address several fundamental problems related to farthest neighbors. Given a subset S ⊆ P , an L 1 geodesic farthest neighbor of p ∈ P from S is one that maximizes the length of L 1 shortest path from p in P. Our list of problems include: computing the diameter, radius, center, farthest-neighbor Voronoi diagram, and two-center of S under the L 1 geodesic distance. We show that all these problems can be solved in linear or near-linear time based on our new observations on farthest neighbors and extreme points. Among them, the key observation shows that there are at most four extreme points of any compact subset S ⊆ P with respect to the L 1 geodesic distance after removing redundancy. [ABSTRACT FROM AUTHOR]

We show that $$\mathbb {R}^3$$ can be packed at a density of $$0.222\ldots $$ with tori whose minor radius goes to zero. Furthermore, we show that the same torus arrangement yields an asymptotically optimal number of pairwise-linked tori. [ABSTRACT FROM AUTHOR]

Let Ω be a disk of radius R in the plane. A set F of unit disks contained in Ω forms a maximal packing if the unit disks are pairwise interior-disjoint and the set is maximal, i.e., it is not possible to add another disk to F while maintaining the packing property. A point p is hidden within the 'forest' defined by F if any ray with apex p intersects some disk of F, that is, a person standing at p can hide without being seen from outside the forest. We show that if the radius R of Ω is large enough, one can find a hidden point for any maximal packing of unit disks in Ω. This proves a conjecture of Joseph Mitchell. We also present an O( nlog n)-time algorithm that, given a forest with n (not necessarily congruent) disks, computes the boundary illumination map of all disks in the forest. [ABSTRACT FROM AUTHOR]

Abstract   Given a sphere of any radius r in an n-dimensional Euclidean space, we study the coverings of this sphere with solid spheres of radius one. Our goal is to design a covering of the lowest covering density, which defines the average number of solid spheres covering a point in a bigger sphere. For growing dimension n, we design a covering that gives the covering density of order (nln n)/2 for a sphere of any radius r>1 and a complete Euclidean space. This new upper bound reduces two times the order nln n established in the classic Rogers bound. [ABSTRACT FROM AUTHOR]

Abstract??In this note we prove that there exists a constant ? such that any planar graph G of diameter ? 2R can be covered with at most ? balls of radius R, a result conjectured by Gavoille, Peleg, Raspaud, and Sopena in 2001. [ABSTRACT FROM AUTHOR]

In 1945, A.W. Goodman and R.E. Goodman proved the following conjecture by P. Erdős: Given a family of (round) disks of radii r1, …, rn in the plane, it is always possible to cover them by a disk of radius R=∑ri, provided they cannot be separated into two subfamilies by a straight line disjoint from the disks. In this note we show that essentially the same idea may work for different analogues and generalizations of their result. In particular, we prove the following: Given a family of positive homothetic copies of a fixed convex body K⊂Rd with homothety coefficients τ1,…,τn>0, it is always possible to cover them by a translate of d+12(∑τi)K, provided they cannot be separated into two subfamilies by a hyperplane disjoint from the homothets. [ABSTRACT FROM AUTHOR]

We consider packings of congruent circles on a square flat torus, i.e., periodic (w.r.t. a square lattice) planar circle packings, with the maximal circle radius. This problem is interesting due to a practical reason-the problem of 'super resolution of images.' We have found optimal arrangements for $$N=6$$ , 7 and 8 circles. Surprisingly, for the case $$N=7$$ there are three different optimal arrangements. Our proof is based on a computer enumeration of toroidal irreducible contact graphs. [ABSTRACT FROM AUTHOR]

A diameter graph in $${\mathbb R}^d$$ is a graph whose set of vertices is a finite subset of $${\mathbb R}^d$$ and whose set of edges is formed by pairs of vertices that are at diameter apart. This paper is devoted to the study of different extremal properties of diameter graphs in $${\mathbb R}^4$$ and on a three-dimensional sphere. We prove an analog of Vázsonyi's and Borsuk's conjecture for diameter graphs on a three-dimensional sphere with radius greater than $$1/\sqrt{2}$$ . We prove Schur's conjecture for diameter graphs in $${\mathbb R}^4.$$ We also establish the maximum number of triangles a diameter graph in $${\mathbb R}^4$$ can have, showing that the extremum is attained only on specific Lenz configurations. [ABSTRACT FROM AUTHOR]

This paper deals with the containment problem under homothetics which has the minimal enclosing ball (MEB) problem as a prominent representative. We connect the problem to results in classic convex geometry and introduce a new series of radii, which we call core-radii. For the MEB problem, these radii have already been considered from a different point of view and sharp inequalities between them are known. In this paper sharp inequalities between core-radii for general containment under homothetics are obtained. Moreover, the presented inequalities are used to derive sharp upper bounds on the size of core-sets for containment under homothetics. In the MEB case, this yields a tight (dimension-independent) bound for the size of such core-sets. In the general case, we show that there are core-sets of size linear in the dimension and that this bound stays sharp even if the container is required to be symmetric. [ABSTRACT FROM AUTHOR]

We consider the offset-deconstruction problem: Given a polygonal shape Q with n vertices, can it be expressed, up to a tolerance ε in Hausdorff distance, as the Minkowski sum of another polygonal shape P with a disk of fixed radius? If it does, we also seek a preferably simple-looking solution P; then, P's offset constitutes an accurate, vertex-reduced, and smoothened approximation of Q. We give an O( nlog n)-time exact decision algorithm that handles any polygonal shape, assuming the real-RAM model of computation. A variant of the algorithm, which we have implemented using the cgal library, is based on rational arithmetic and answers the same deconstruction problem up to an uncertainty parameter δ; its running time additionally depends on δ. If the input shape is found to be approximable, this algorithm also computes an approximate solution for the problem. It also allows us to solve parameter-optimization problems induced by the offset-deconstruction problem. For convex shapes, the complexity of the exact decision algorithm drops to O( n), which is also the time required to compute a solution P with at most one more vertex than a vertex-minimal one. [ABSTRACT FROM AUTHOR]

A polygon is said to be simple if the only points of the plane belonging to two of its edges are its vertices. We answer the question of finding, for a given integer n, a simple n-sided polygon contained in a disk of radius 1 that has the longest perimeter. When n is even, the optimal solution is arbitrarily close to a line segment of length 2 n. When n is odd, the optimal solution is arbitrarily close to an isosceles triangle. [ABSTRACT FROM AUTHOR]

Abstract  We study two notions. One is that of spindle convexity. A set of circumradius not greater than one is spindle convex if, for any pair of its points, it contains every short circular arc of radius at least one, connecting them. The other objects of study are bodies obtained as intersections of finitely many balls of the same radius, called ball-polyhedra. We find analogues of several results on convex polyhedral sets for ball-polyhedra. [ABSTRACT FROM AUTHOR]

Abstract??What is the minimum radiusof a spherical container inthat can hold n unit balls, and how must the balls be arranged in such a container? This question is equivalent to: How should n points be selected in the unit ball inso that the minimum distance between any two of them be as large as possible, and what is that distance? Davenport and Hajos, and, independently, Rankin, proved that if F is a set of d + 2 points in the unit ball in, then two of the points in F are at a distance of at mostfrom each other. Rankin proved also that if F consists of 2d points in the ball such that the distance between any two of them is at least, then their configuration is unique up to an isometry, namely the points must be the vertices of a regular d-dimensional crosspolytope inscribed in the ball. However, if d + 2 ? n ? 2d - 1, then the optimal arrangements of n points (i.e., those that maximize the smallest distance between them) are not unique. Here we generalize the results from Davenport and Hajos and Rankin by describing all possible optimal configurations, unique or not, of n = d + 2, d + 3,..., 2d points. [ABSTRACT FROM AUTHOR]

Suppose you are given a collection of discs of radius e and you are asked to place them in the plane so that any straight line that crosses the unit disc will hit one or more of them. Furthermore, you are asked to do this with as few discs as possible. How many do you need? This problem can be cast as an (infinite) integer programming (IP) problem. Often, for such problems, one can obtain a lower bound by considering the linear programming (LP) relaxation. For the above problem the relaxed problem allows discs with positive weight, requires that each line crossing the unit disc accumulates weight 1 from discs it hits, and asks that the total weight be minimized. This LP problem has a simple asymptotically optimal solution with total weight 1/e. Clearly, this is the best possible since blocking all the lines in a single direction requires total weight 1/e. Although it is quite intuitive that the IP version cannot have such a ?perfect? solution, achieving the lower bound, it is surprisingly difficult to obtain a better lower bound. Here we improve the lower bound to 1.001/e. The true answer is probably considerably larger. We believe this is the first example of a problem in this class where it has been proven that the LP and IP versions have different answers. [ABSTRACT FROM AUTHOR]

We show that there is a query expressible in first-order logic over the reals that returns, on any given semi-algebraic set A, for every point, a radius around which A is conical in every small enough box. We obtain this result by combining results from differential topology and real algebraic geometry, with recent algorithmic results by Rannou. [ABSTRACT FROM AUTHOR]