Strict Kneser-Poulsen conjecture for large radii

In this paper we prove the Kneser-Poulsen conjecture for the case of large radii. Namely, if a finite number of points in Euclidean space $E^n$ is rearranged so that the distance between each pair of points does not decrease, then there exists a positive number $r_0$ that depends on the rearrangement of the points, such that if we consider $n$-dimensional balls of radius $r>r_0$ with centers at these points, then the volume of the union (intersection) of the balls before the rearrangement is not less (not greater) than the volume of the union (intersection) after the rearrangement. Moreover, the inequality is strict whenever the new point set is not congruent to the original one. Also under the same conditions we prove a similar result about surface volumes instead of volumes. In order to prove the above mentioned results we use ideas from tensegrity theory to strengthen the theorem of Sudakov, R. Alexander and Capoyleas and Pach, which says that the mean width of the convex hull of a finite number of points does not decrease after an expansive rearrangement of those points. In this paper we show that the mean width increases strictly, unless the expansive rearrangement was a congruence. We also show that if the configuration of centers of the balls is fixed and the volume of the intersection of the balls is considered as a function of the radius $r$, then the second highest term in the asymptotic expansion of this function is equal to $-M_n r^{n-1}$, where $M_n$ is the mean width of the convex hall of the centers. This theorem was conjectured by Balazs Csikos in 2009.Comment: 14 pages, 1 figur

On the Volume of Boolean expressions of Large Congruent Balls

We consider the volume of a Boolean expression of some congruent balls about a given system of centers in the d-dimensional Euclidean space. When the radius r of the balls is large, this volume can be approximated by a polynomial of r, which will be computed up to an O(r^{d−3}) error term. We study how the top coefficients of this polynomial depend on the set of the centers. It is known that in the case of the union of the balls, the top coefficients are some constant multiples of the intrinsic volumes of the convex hull of the centers. Thus, the coefficients in the general case lead to generalizations of the intrinsic volumes, in particular, to a generalization of the mean width of a set. Some known results on the mean width, along with the theorem on its monotonicity under contractions are extended to the "Boolean analogues" of the mean width

Computing the Volume of a Union of Balls: a Certified Algorithm

Balls and spheres are amongst the simplest 3D modeling primitives, and computing the volume of a union of balls is an elementary problem. Although a number of strategies addressing this problem have been investigated in several communities, we are not aware of any robust algorithm, and present the first such algorithm. Our calculation relies on the decomposition of the volume of the union into convex regions, namely the restrictions of the balls to their regions in the power diagram. Theoretically, we establish a formula for the volume of a restriction, based on Gauss' divergence theorem. The proof being constructive, we develop the associated algorithm. On the implementation side, we carefully analyse the predicates and constructions involved in the volume calculation, and present a certified implementation relying on interval arithmetic. The result is certified in the sense that the exact volume belongs to the interval computed using the interval arithmetic. Experimental results are presented on hand-crafted models presenting various difficulties, as well as on the 58,898 models found in the 2009-07-10 release of the Protein Data Bank

Multiple covers with balls I: Inclusion–exclusion

Inclusion–exclusion is an effective method for computing the volume of a union of measurable sets. We extend it to multiple coverings, proving short inclusion–exclusion formulas for the subset of Rn covered by at least k balls in a finite set. We implement two of the formulas in dimension n=3 and report on results obtained with our software

Sampling Conditions for Conforming Voronoi Meshing by the VoroCrust Algorithm

We study the problem of decomposing a volume bounded by a smooth surface into a collection of Voronoi cells. Unlike the dual problem of conforming Delaunay meshing, a principled solution to this problem for generic smooth surfaces remained elusive. VoroCrust leverages ideas from alpha-shapes and the power crust algorithm to produce unweighted Voronoi cells conforming to the surface, yielding the first provably-correct algorithm for this problem. Given an epsilon-sample on the bounding surface, with a weak sigma-sparsity condition, we work with the balls of radius delta times the local feature size centered at each sample. The corners of this union of balls are the Voronoi sites, on both sides of the surface. The facets common to cells on opposite sides reconstruct the surface. For appropriate values of epsilon, sigma and delta, we prove that the surface reconstruction is isotopic to the bounding surface. With the surface protected, the enclosed volume can be further decomposed into an isotopic volume mesh of fat Voronoi cells by generating a bounded number of sites in its interior. Compared to state-of-the-art methods based on clipping, VoroCrust cells are full Voronoi cells, with convexity and fatness guarantees. Compared to the power crust algorithm, VoroCrust cells are not filtered, are unweighted, and offer greater flexibility in meshing the enclosed volume by either structured grids or random samples

Greedy Geometric Optimization Algorithms for Collection of Balls

Modeling 3D objects with balls is routine for two reasons: on the one hand, the medial axis transform allows representing a solid object as a union of medial balls; on the other hand, selected shapes, and molecules in particular, are naturally represented by collections of balls. Yet, the problem of choosing which balls are best suited to approximate a given shape is a non trivial one. This paper addresses two problems in this realm. The first one, conformational diversity selection, consists of choosing $k$ molecular conformations amidst $n$, so as to maximize the geometric diversity of the $k$ conformers. The second one, inner approximation, consists of approximating a molecule of $n$ balls with $k$ balls. On the theoretical side, we demonstrate that for both problems, a geometric generalization of max $k$-cover applies, with weights depending on the cells of a surface or volumetric arrangement. Tackling these problems with greedy strategies, it is shown that the $1-1/e$ bound known in combinatorial optimization applies in some cases but not all. On the applied side, we present a robust and effective implementation of the greedy algorithm for the inner approximation problem, which incorporates the calculation of the exact Delaunay triangulation of a points whose coordinates are degree two algebraic number, of the medial axis of a union of balls, and of a certified estimate of the volume of a union of balls. In particular, we show that the inner approximation of complex molecules yields accurate coarse-grain models with a number of balls 100 times smaller than the number of atoms, a key requirement to simulate crowded protein environments.Les boules jouent un rôle central en modélisation géométrique pour deux raisons: d'une part la transformée associée à l'axe médian permet de représenter un objet solide comme une union in nie de boules; d'autre part, certaines formes, et les modèles moléculaires de van der Waals en particulier, sont dé nies par une union de boules. Néanmoins, la question de savoir quel ensemble de boules utiliser pour approximer une forme est non trivial, de telle sorte que ce travail aborde deux problèmes liés. Pour les présenter, par conformation moléculaire, nous entendons un modèle dé ni par un ensemble ni de boules. La premier problème, ou selection de diversité géométrique, consiste à choisir k conformations moléculaires parmi n, de façon à maximiser la diversité de l'ensemble choisi. Le second, ou approximation par défaut, consiste à approximer une molécule de n boules par k < n boules. Du point de vue théorique, nous montrons que les deux problèmes peuvent être traités avec une variante géométrique de max k-cover, les poids dépendant de la géométrie d'un arrangement surfacique ou volumique de sphères. La résolution de ces problèmes par un algorithme glouton permet d'avoir un facteur d'approximation borné inférieurement par 1 1=e dans certains cas. D'un point de vue appliqué, nous présentons une implémentation robuste de l'algorithme glouton pour l'approximation par défaut, laquelle incorpore (i) le calcul exact d'une triangulation de Delaunay dont les points ont des coordonnées qui sont des nombres algébriques de degré deux, (ii) le calcul exact de l'axe médian d'une union de boules, et (iii) une approximation certi ée du volume d'une union de boules. En particulier, nous montrons que des approximations précises de modèles moléculaires peuvent être obtenues en utilisant un nombre de boules 100 fois inférieur au nombre d'atomes, une propriété particulièrement séduisante pour la simulation d'environnement protéique dense