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HellyType Theorems and Geometric Transversals
 Handbook of Discrete and Computational Geometry, chapter 4
, 1997
"... INTRODUCTION A geometric transversal is an affine subspace of R d , such as a point, line, plane or hyperplane, which intersects every member of a family of convex sets. Eduard Helly's celebrated theorem gives conditions for the members of a family of convex sets to have a point in common, i.e., ..."
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Cited by 35 (3 self)
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INTRODUCTION A geometric transversal is an affine subspace of R d , such as a point, line, plane or hyperplane, which intersects every member of a family of convex sets. Eduard Helly's celebrated theorem gives conditions for the members of a family of convex sets to have a point in common, i.e., a point transversal. In Section 1 we highlight some of the more notable theorems related to Helly's Theorem and point transversals. Section 2 is devoted to geometric transversal theory. 4.1 HELLYTYPE THEOREMS In 1913, Eduard Helly proved the following theorem: Theorem 1 (Helly's Theorem) Let A be a finite family of at least d + 1 convex sets in R d . If every d + 1 members of A have a point in common, then there is a point common to all the members of A. The theorem also holds for infinite families
Some discrete properties of the space of line transversals to disjoint balls
 In NonLinear Computational Geometry
, 2008
"... Abstract. Attempts to generalize Helly’s theorem to sets of lines intersecting convex sets led to a series of results relating the geometry of a family of sets in R d to the structure of the space of lines intersecting all of its members. We review recent progress in the special case of disjoint Euc ..."
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Cited by 6 (5 self)
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Abstract. Attempts to generalize Helly’s theorem to sets of lines intersecting convex sets led to a series of results relating the geometry of a family of sets in R d to the structure of the space of lines intersecting all of its members. We review recent progress in the special case of disjoint Euclidean balls in R d, more precisely the interrelated notions of cone of directions, geometric permutations and Hellytype theorems, and discuss some algorithmic applications. Key words. Geometric transversal, Helly’s theorem, line, sphere, geometric permutation, cone of directions. 1. Introduction. Lines
Line problems in nonlinear computational geometry
 in Computational Geometry  Twenty Years
"... Abstract. We first review some topics in the classical computational geometry of lines, in particular the O(n 3+ǫ) bounds for the combinatorial complexity of the set of lines in R 3 interacting with n objects of fixed description complexity. The main part of this survey is recent work on a core alge ..."
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Cited by 4 (0 self)
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Abstract. We first review some topics in the classical computational geometry of lines, in particular the O(n 3+ǫ) bounds for the combinatorial complexity of the set of lines in R 3 interacting with n objects of fixed description complexity. The main part of this survey is recent work on a core algebraic problem—studying the lines tangent to k spheres that also meet 4−k fixed lines. 1.
On the Connected Components of the Space of Line Transversals to a Family of Convex Sets
 Discrete Comput. Geom
, 1997
"... Let L be the space of line transversals to a finite family of pairwise disjoint compact convex sets in R 3 . We prove that each connected component of L can itself be represented as the space of transversals to some finite family of pairwise disjoint compact convex sets. Introduction Let A be a f ..."
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Cited by 3 (2 self)
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Let L be the space of line transversals to a finite family of pairwise disjoint compact convex sets in R 3 . We prove that each connected component of L can itself be represented as the space of transversals to some finite family of pairwise disjoint compact convex sets. Introduction Let A be a family of convex sets in R d . A ktransversal for A is a kflat, an affine subspace of dimension k, that intersects every member of A. (For basic facts about transversal theory, see [1] or [3].) Let A be the set of all k transversals of A, considered as a subspace of the affine Grassmannian G 0 k;d of all kflats in R d . We are interested in studying the connected components of this space A and, in particular, in the question of whether each of these connected components can be represented as the space of ktransversals of some other family B of convex sets. City College, City University of New York, New York, NY 10031, U.S.A. (jegcc@cunyvm.cuny.edu). Supported in part by...
convexity, frame convexity, and a theorem of Santaló
 Advances in Geometry
"... Abstract. In 1940, Luis Santaló proved a Hellytype theorem for line transversals to boxes in R d. An analysis of his proof reveals a convexity structure for ascending lines in R d that is isomorphic to the ordinary notion of convexity in a convex subset of R 2d−2. This isomorphism is through a Crem ..."
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Cited by 2 (0 self)
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Abstract. In 1940, Luis Santaló proved a Hellytype theorem for line transversals to boxes in R d. An analysis of his proof reveals a convexity structure for ascending lines in R d that is isomorphic to the ordinary notion of convexity in a convex subset of R 2d−2. This isomorphism is through a Cremona transformation on the Grassmannian of lines in P d, which enables a precise description of the convex hull and affine span of up to d ascending lines: the lines in such an affine span turn out to be the rulings of certain classical determinantal varieties. Finally, we relate Cremona convexity to a new convexity structure that we call frame convexity, which extends to arbitrarydimensional flats in R d.
KTransversals of Parallel Convex Sets
, 1996
"... R d can be divided into a union of parallel (d\Gammak)flats of the form x 1 = g 1 ; x 2 = g 2 ; : : : x k = g k , where the g i are constant. Let C be a family of parallel (d \Gamma k)dimensional convex sets, meaning that each is contained in one of the above parallel (d \Gamma k)flats. We giv ..."
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R d can be divided into a union of parallel (d\Gammak)flats of the form x 1 = g 1 ; x 2 = g 2 ; : : : x k = g k , where the g i are constant. Let C be a family of parallel (d \Gamma k)dimensional convex sets, meaning that each is contained in one of the above parallel (d \Gamma k)flats. We give a parameterization of the set of kflats in R d , such that the set of kflats which intersect, in a point, any set c 2 C, is convex. Parameterizing the lines in R 3 through horizontal convex sets as convex sets has applications to medical imaging, and interesting connections with recent work on light field rendering in computer graphics. The general case is useful for fitting kflats to points in R d . The following easy reduction is well known. Let C be a finite set of parallel line segments in R d . We want to find a (d \Gamma 1)transversal for C, that is, a hyperplane intersecting every segment in C. Such a hyperplane has to pass below the upper endpoint of each segment and ...
Helly numbers of acyclic families
, 2011
"... The Helly number of a family of sets with empty intersection is the size of its largest inclusionwise minimal subfamily with empty intersection. Let F be a finite family of open subsets of an arbitrary locally arcwise connected topological space Γ. Assume that for every subfamily G ⊆ F the inters ..."
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The Helly number of a family of sets with empty intersection is the size of its largest inclusionwise minimal subfamily with empty intersection. Let F be a finite family of open subsets of an arbitrary locally arcwise connected topological space Γ. Assume that for every subfamily G ⊆ F the intersection of the elements of G has at most r connected components, each of which is a Qhomology cell. We show that the Helly number of F is at most r(dΓ + 1), where dΓ is the smallest integer j such that every open set of Γ has trivial Qhomology in dimension j and higher. (In particular dRd = d). This bound is best possible. We prove, in fact, a stronger theorem where small subfamilies may have more than r connected components, each possibly with nontrivial homology in low dimension. As an application, we obtain several explicit bounds on Helly numbers in geometric transversal theory for which only ad hoc geometric proofs were previously known; in certain cases, the bound we obtain is better than what was previously known. 1
Motion Coordination for . . . WITH VISIBILITY SENSORS
, 2007
"... The subject of this dissertation is motion coordination for mobile robotic networks with visibility sensors. Such networks consist of robotic agents equipped with sensors that can measure distances to the environment boundary and to other agents within line of sight. We look at two fundamental coord ..."
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The subject of this dissertation is motion coordination for mobile robotic networks with visibility sensors. Such networks consist of robotic agents equipped with sensors that can measure distances to the environment boundary and to other agents within line of sight. We look at two fundamental coordination problems: (i) deploying over an unknown nonconvex environment to achieve complete visibility, and (ii) gathering all agents initially scattered over the environment at a single location. As a special case of problem (i), we first address the problem of optimally locating a single robotic agent in a nonconvex environment. The agent is modeled as a point mass with continuous firstorder dynamics. We propose a nonsmooth gradient algorithm for the problem of maximizing the area of the region visible to the observer in a nonselfintersecting nonconvex polygon. First, we show that the visible area is almost everywhere a locally Lipschitz function of the observer location. Second, we provide a novel version of the LaSalle Invariance Principle for discontinuous vector fields and for Lyapunov functions with a finite number of discontinuities. Finally, we establish the asymptotic convergence properties of the nonsmooth gradient algorithm and we illustrate numerically its performance. Second, we address problem (i) by proposing a novel algorithm to the deploy a group of robotic