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Voronoi Diagrams
 Handbook of Computational Geometry
"... Voronoi diagrams can also be thought of as lower envelopes, in the sense mentioned at the beginning of this subsection. Namely, for each point x not situated on a bisecting curve, the relation p x q defines a total ordering on S. If we construct a set of surfaces H p , p S,in3space such t ..."
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Cited by 143 (19 self)
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Voronoi diagrams can also be thought of as lower envelopes, in the sense mentioned at the beginning of this subsection. Namely, for each point x not situated on a bisecting curve, the relation p x q defines a total ordering on S. If we construct a set of surfaces H p , p S,in3space such that H p is below H q i# p x q holds, then the projection of their lower envelope equals the abstract Voronoi diagram.
The nonlinear geometry of linear programming IV. Hilbert geometry, in preparation
"... This series of papers studies a geometric structure underlying Karmarkar’s projective scaling algorithm for solving linear programming problems. A basic feature of the projective scaling algorithm is a vector field depending on the objective function which is defined on the interior of the polytope ..."
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Cited by 67 (0 self)
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This series of papers studies a geometric structure underlying Karmarkar’s projective scaling algorithm for solving linear programming problems. A basic feature of the projective scaling algorithm is a vector field depending on the objective function which is defined on the interior of the polytope of feasible solutions of the linear program. The geometric structure we study is the set of trajectories obtained by integrating this vector field, which we call Ptrajectories. In order to study Ptrajectories we also study a related vector field on the linear programming polytope, which we call the affine scaling vector field, and its associated trajectories, called Atrajectories. The affine scaling vector field is associated to another linear programming algorithm, the affine scaling algorithm. These affine and projective scaling vector fields are each defined for liner programs of a special form, called strict standard form and canonical form, respectively. This paper defines and presents basic properties of Ptrajectories and Atrajectories. It reviews the projective and affine scaling algorithms, defines the projective and affine scaling vector fields, and gives differential equations for Ptrajectories and Atrajectories. It presents Karmarkar’s interpretation of Atrajectories as steepest descent paths of the objective function 〈c, x 〉 with respect to the Riemannian _ dx
Applications of convex analysis to multidimensional scaling
 Recent Developments in Statistics
, 1977
"... Abstract. In this paper we discuss the convergence of an algorithm for metric and nonmetric multidimensional scaling that is very similar to the Cmatrix algorithm of Guttman. The paper improves some earlier results in two respects. In the first place the analysis is extended to cover general Minkov ..."
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Cited by 52 (5 self)
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Abstract. In this paper we discuss the convergence of an algorithm for metric and nonmetric multidimensional scaling that is very similar to the Cmatrix algorithm of Guttman. The paper improves some earlier results in two respects. In the first place the analysis is extended to cover general Minkovski metrics, in the second place a more elementary proof of convergence based on results of Robert is presented. This paper was originally presented at the European Meeting of Statisticians,
Cut locus and medial axis in global shape interrogation and representation
 MIT Design Laboratory Memorandum 922 and MIT Sea Grant Report
, 1992
"... The cut locus CA of a closed set A in the Euclidean space E is defined as the closure of the set containing all points p which have at least two shortest paths to A. We present a theorem stating that the complement of the cut locus i.e. E\(CA∪A) is the maximal open set in (E\A) where the distance fu ..."
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Cited by 35 (1 self)
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The cut locus CA of a closed set A in the Euclidean space E is defined as the closure of the set containing all points p which have at least two shortest paths to A. We present a theorem stating that the complement of the cut locus i.e. E\(CA∪A) is the maximal open set in (E\A) where the distance function with respect to the set A is continuously differentiable. This theorem includes also the result that this distance function has a locally Lipschitz continuous gradient on (E\A). The medial axis of a solid D in E is defined as the union of all centers of all maximal discs which fit in this domain. We assume in the medial axis case that D is closed and that the boundary ∂D of D is a topological (not necessarily connected) hypersurface of E. Under these assumptions we prove that the medial axis of D equals that part of the cut locus of ∂D which is contained in D. We prove that the medial axis has the same homotopy type as its reference solid if the solid’s boundary surface fulfills certain regularity requirements. We also show that the medial axis with its related distance function can be be used to reconstruct its reference solid. We prove that the cut locus of a solid’s boundary is nowhere dense in the Euclidean space if the solid’s boundary meets certain regularity requirements. We show that the cut locus concept offers a common frame work lucidly unifying different concepts such as Voronoi diagrams, medial axes and equidistantial point sets. In this context we prove that the equidistantial set of two disjoint
Astrogeometry, Error Estimation, and Other Applications of SetValued Analysis
 ACM SIGNUM Newsletter
, 1996
"... In many reallife application problems, we are interested in numbers, namely, in the numerical values of the physical quantities. There are, however, at least two classes of problems, in which we are actually interested in sets: ffl In image processing (e.g., in astronomy), the desired blackand ..."
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Cited by 27 (26 self)
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In many reallife application problems, we are interested in numbers, namely, in the numerical values of the physical quantities. There are, however, at least two classes of problems, in which we are actually interested in sets: ffl In image processing (e.g., in astronomy), the desired blackandwhite image is, from the mathematical viewpoint, a set.
Graphs of some CAT(0) complexes
 Adv. Appl. Math
, 1998
"... In this note, we characterize the graphs (1skeletons) of some piecewise Euclidean simplicial and cubical complexes having nonpositive curvature in the sense of Gromov’s CAT(0) inequality. Each such cell complex K is simply connected and obeys a certain flag condition. It turns out that if, in addit ..."
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Cited by 26 (12 self)
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In this note, we characterize the graphs (1skeletons) of some piecewise Euclidean simplicial and cubical complexes having nonpositive curvature in the sense of Gromov’s CAT(0) inequality. Each such cell complex K is simply connected and obeys a certain flag condition. It turns out that if, in addition, all maximal cells are either regular Euclidean cubes or right Euclidean triangles glued in a special way, then the underlying graph G�K � is either a median graph or a hereditary modular graph without two forbidden induced subgraphs. We also characterize the simplicial complexes arising from bridged graphs, a class of graphs whose metric enjoys one of the basic properties of CAT(0) spaces. Additionally, we show that the graphs of all these complexes and some more general classes of graphs have geodesic combings and bicombings verifying the 1 or 2fellow traveler property. © 2000 Academic Press 1.
The Local Structure of Length Spaces With Curvature Bounded Above
, 1998
"... We show that a number of different notions of dimension coincide for length spaces with curvature bounded above. We then apply this result, showing that if X is a locally compact CAT (0) space with cocompact isometry group, then the dimension of the Tits boundary and the asymptotic cone(s) of X are ..."
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Cited by 22 (0 self)
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We show that a number of different notions of dimension coincide for length spaces with curvature bounded above. We then apply this result, showing that if X is a locally compact CAT (0) space with cocompact isometry group, then the dimension of the Tits boundary and the asymptotic cone(s) of X are determined by the maximal dimension of a flat in X. 1
Coxeter groups, Salem numbers and the Hilbert metric
, 2001
"... this paper we prove a similar result for loops in the fundamental polyhedron of a Coxeter group W , and use it to study the spectral radius (w), w 2 W for the geometric action of W . In particular we prove: Theorem 1.1 Let (W; S) be a Coxeter system and let w 2 W . Then either (w) = 1 or (w) Le ..."
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Cited by 21 (5 self)
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this paper we prove a similar result for loops in the fundamental polyhedron of a Coxeter group W , and use it to study the spectral radius (w), w 2 W for the geometric action of W . In particular we prove: Theorem 1.1 Let (W; S) be a Coxeter system and let w 2 W . Then either (w) = 1 or (w) Lehmer 1:1762808. Here Lehmer denotes Lehmer's number, a root of the polynomial 1 + x \Gamma x 3 \Gamma x 4 \Gamma x 5 \Gamma x 6 \Gamma x 7 + x 9 + x 10 (1.1) and the smallest known Salem number. Billiards. Recall that a Coxeter system (W; S) is a group W with a finite generating set S = fs 1 ; : : : ; s n g, subject only to the relations (s i s j ) m ij = 1, where m ii = 1 and m ij 2 for i 6= j. The permuted products s oe1 s oe2 \Delta \Delta \Delta s oen 2 W; oe 2 S n ; are the Coxeter elements of (W; S). We say w 2 W is essential if it is not conjugate into any subgroup W I ae W generated by a proper subset I ae S. The Coxeter group W acts naturally by reflections on V
Graphs With Connected Medians
, 2002
"... The median set of a graph G with weighted vertices comprises the vertices minimizing the average weighted distance to the vertices of G: We characterize the graphs in which, with respect to any non negative vertex weights, median sets always induce connected subgraphs. The characteristic condition ..."
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Cited by 17 (3 self)
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The median set of a graph G with weighted vertices comprises the vertices minimizing the average weighted distance to the vertices of G: We characterize the graphs in which, with respect to any non negative vertex weights, median sets always induce connected subgraphs. The characteristic conditions can be tested in polynomial time (by employing linear programming) and are immediately verified for a number of specific graph classes.