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Line Transversals of Convex Polyhedra in R 3∗
, 2008
"... We establish a bound of O(n 2 k 1+ε), for any ε> 0, on the combinatorial complexity of the set T of line transversals of a collection P of k convex polyhedra in R 3 with a total of n facets, and present a randomized algorithm which computes the boundary of T in comparable expected time. Thus, whe ..."
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We establish a bound of O(n 2 k 1+ε), for any ε> 0, on the combinatorial complexity of the set T of line transversals of a collection P of k convex polyhedra in R 3 with a total of n facets, and present a randomized algorithm which computes the boundary of T in comparable expected time. Thus
Line Transversals of Convex Polyhedra in R³
, 2008
"... We establish a bound of O(n 2 k 1+ε), for any ε> 0, on the combinatorial complexity of the set T of line transversals of a collection P of k convex polyhedra in R 3 with a total of n facets, and present a randomized algorithm which computes the boundary of T in comparable expected time. Thus, whe ..."
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We establish a bound of O(n 2 k 1+ε), for any ε> 0, on the combinatorial complexity of the set T of line transversals of a collection P of k convex polyhedra in R 3 with a total of n facets, and present a randomized algorithm which computes the boundary of T in comparable expected time. Thus
Submodular functions, matroids and certain polyhedra
, 2003
"... The viewpoint of the subject of matroids, and related areas of lattice theory, has always been, in one way or another, abstraction of algebraic dependence or, equivalently, abstraction of the incidence relations in geometric representations of algebra. Often one of the main derived facts is that all ..."
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of linear programming. It turns out to be useful to regard “pure matroid theory”, which is only incidentally related to the aspects of algebra which it abstracts, as the study of certain classes of convex polyhedra. (1) A matroid M = (E,F) can be defined as a finite set E and a nonempty family F of so
Transverse cellular mappings of polyhedra
 Trans. Amer. Math. Soc
, 1972
"... ABSTRACT. We generalize Marshall Cohen's notion of transverse cellular map to the polyhedral category. They are described by the following: Proposition. Let f: K — * L be a proper simplicial map of locally finite simplicial complexes. The following are equivalent: (1) The dual cells of the map ..."
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ABSTRACT. We generalize Marshall Cohen's notion of transverse cellular map to the polyhedral category. They are described by the following: Proposition. Let f: K — * L be a proper simplicial map of locally finite simplicial complexes. The following are equivalent: (1) The dual cells of the map
On the infinitesimal rigidity of weakly convex polyhedra
, 606
"... The main motivation here is a question: whether any polyhedron which can be subdivided into convex pieces without adding a vertex, and which has the same vertices as a convex polyhedron, is infinitesimally rigid. We prove that it is indeed the case for two classes of polyhedra: those obtained from a ..."
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The main motivation here is a question: whether any polyhedron which can be subdivided into convex pieces without adding a vertex, and which has the same vertices as a convex polyhedron, is infinitesimally rigid. We prove that it is indeed the case for two classes of polyhedra: those obtained from
Convex polyhedra in R³ spanning . . .
"... We construct nvertex convex polyhedra with the property stated in the title ..."
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We construct nvertex convex polyhedra with the property stated in the title
An Illumination Problem for Convex Polyhedra
"... Consider a convex polytope P in the ddimensional Euclidean space. We say that a vertex v of P illuminates a point u 2 E d lying outside P if the line segment uv does not intersect the interior of P . Furthermore, we say that the vertices v 1 ; v 2 ; : : : ; v l of P illuminate the entire exterior ..."
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Consider a convex polytope P in the ddimensional Euclidean space. We say that a vertex v of P illuminates a point u 2 E d lying outside P if the line segment uv does not intersect the interior of P . Furthermore, we say that the vertices v 1 ; v 2 ; : : : ; v l of P illuminate the entire
Blaschke Addition and Convex Polyhedra
"... Abstract. This is an extended version of a talk on October 4, 2004 at the research seminar “Differential geometry and applications ” (headed by Academician A. T. Fomenko) at Moscow State University. The paper contains an overview of available (but far from wellknown) results about the Blaschke addi ..."
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addition of convex bodies, some new theorems on the monotonicity of the volume of convex bodies (in particular, convex polyhedra with parallel faces) as well as description of a software for visualization of polyhedra with prescribed outward normals and face areas. 1. The vector area of the surface of a
Blaschke Addition and Convex Polyhedra
"... This is an extended version of a talk on October 4, 2004 at the research seminar “Differential geometry and applications” (headed by Academician A. T. Fomenko) at Moscow State University. The paper contains an overview of available (but far from wellknown) results about the Blaschke addition of con ..."
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of convex bodies, some new theorems on the monotonicity of the volume of convex bodies (in particular, convex polyhedra with parallel faces) as well as description of a software for visualization of polyhedra with prescribed outward normals and face areas.
On Hamiltonian tetrahedralizations of convex polyhedra
 Proceedings of ISORA’05 (Lecture Notes on Operations Research
, 2005
"... Let Tp denote any tetrahedralization of a convex polyhedron P and let G T be the dual graph of Tp such that each node of G T corresponds to a tetrahedron of Tp and two nodes are connected by an edge in G T if and only if the two corresponding tetrahedra share a common facet in Tp. Tp is called a Ham ..."
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method does not yield a Hamiltonian tetrahedralization, here the pulling method is the simplest method to ensure a linearsize decomposition and is one of the most commonly used tetrahedralization methods for convex polyhedra. Furthermore, we can construct a convex polyhedron with n vertices
Results 1  10
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426,075