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Topic: Convex polytope

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	Polytope - Wikipedia, the free encyclopedia
		One special kind of polytope is a convex polytope, which is the convex hull of a finite set of points.
		The faces of a convex polytope thus form a lattice called its face lattice, where the subset relation is defined between basis hyperplanes.
		Polytopes may be regarded as a tessellation of some sort of the manifold of their surface.
	en.wikipedia.org /wiki/Polytope (977 words)

	Cross-polytope - Wikipedia, the free encyclopedia
		In geometry, a cross-polytope, or orthoplex, is a regular, convex polytope that exists in any number of dimensions.
		The cross-polytope is the convex hull of its vertices.
		In 3-dimensions it is an octahedron—one of the five regular polyhedra known as the Platonic solids.
	en.wikipedia.org /wiki/Cross-polytope (621 words)

	Peek terminology
		A sphere is convex, as is a triangle, as is R^n.
		Polytopes are the generalization of polygons and polyhedra: a polygon is a two dimensional polytope; a polyhedron is a three dimensional polytope.
		The dimensionality of a polytope is defined by the dimension of the volume it encloses.
	www.cs.utah.edu /~gk/peek/old/terms.html (934 words)

	[No title]
		"Polytope" is usually a synonym for "convex polytope": the convex hull of a finite set of points, or equivalently, the bounded intersection of a finite number of halfspaces.
		This use of "simple" is really confusing, since "simple polytope" usually means a convex polytope in which every vertex lies on d facets.
		Still other people use "polytope" to mean a bounded convex flat-sided thingie, and "polyhedron" to mean a bounded not-necessarily-convex flat-sided thingie -- so polytopes are convex polyhedra.
	www.math.niu.edu /~rusin/known-math/98/polytope (776 words)

	Unfolding convex polytopes
		An unfolding of a 3-dimensional convex polytope is obtained by cutting the polytope along some of its edges (necessarily a spanning tree of the edge graph) and flattening the boundary of the polytope along the remaining edges.
		Günther Rote recently constructed a family of polytopes whose minimum-perimeter unfoldings, constructed by cutting along the minimum spanning tree of the polytope's edge graph, are nonsimple.
		One of Rote's polytopes and its nonsimple minimum-perimeter unfolding.
	compgeom.cs.uiuc.edu /~jeffe/open/unfold.html (768 words)

	David Bremner's computational convexity bibliography
		An algorithm for enumerating the vertices of a convex polyhedron.
		A pivoting algorithm for convex hulls and vertex enumeration of arrangements and polyhedra.
		This triangulation induces a triangulation of the corresponding convex polytope in the representable case.
	www.cs.unb.ca /profs/bremner/PolytopeBase/biblio/bibmain.html (4008 words)

	Multidimensional Glossary
		As a rule, the facets of a conjugate polytope are conjugates of the facets of the first polytope, and the vertex figures of a conjugate polytope are conjugates of the vertex figures of the first polytope, whenever such conjugacies are apparent.
		In four-space, the rectified demicross polytope belongs to the ico regiment, that is, the regiment whose colonel is the regular icositetrachoron.
		That is, the dual of a stellation of a polytope is a faceting of the dual of the polytope.
	members.aol.com /Polycell/glossary.html (16079 words)

	CPC Licence Alert (Site not responding. Last check: 2007-10-12)
		The convex hull is the minimum volume convex polytope which encloses the set of points.
		This condition is handled by recursively adding and removing facets until the polytope returns to a convex state.
		The final convex polytope, the convex hull, is defined by a set of interlocking facets.
	www.cpc.cs.qub.ac.uk /summaries/ADFS.html (490 words)

	Building convex polytopes
		Any convex 3polytope naturally defines a polyhedral metric; the "distance" between two points is just the length of the shortest path on the polytope's surface.
		A fairly simple theorem of Minkowski states that if you take the normal vectors of the facets of a convex polytope, where the length of the vector is the area of the corresponding facet, then the resulting collection of vectors sum to zero.
		This polytope is unique up to reflections and rotations about the origin, and every combinatorial symmetry of the graph is realized by a symmetry of the polytope.
	compgeom.cs.uiuc.edu /~jeffe/open/makepoly.html (1455 words)

	3D Convex Hulls (Site not responding. Last check: 2007-10-12)
		Both versions accept a range of input iterators defining the set of points whose convex hull is to be computed and a traits class defining the geometric types and predicates used in computing the hull.
		Then the number of points of the convex hull are obtained by counting the number of triangulation vertices incident to the infinite vertex.
		Notice that the vertices incident to the infinite vertex of the triangulation are on the convex hull but it may be that not all of them are vertices of the hull.
	www.cgal.org /Manual/doc_html/cgal_manual/Convex_hull_3/Chapter_main.html (706 words)

	Polytopes (Site not responding. Last check: 2007-10-12)
		Polytope is the general term of the sequence, point, segment, polygon, polyhedron,...
		In four dimensions, there are sixteen regular polytopes: six convex, and ten starry; but in each and every higher space, there are but three: the regular simplex, the cross polytope or orthoplex, and the hypercube.
		A rectified polytope has for vertices the mid-edge points of its parent, while a truncated regular polytope is typically imagined to be truncated by hyperplanes perpendicular to the vectors to its vertices, to just such a depth as would create a regular 2n-gon, from any one of the bounding n-gons.
	users.neworld.net /~rtowle/Polytopes/polytope.html (1636 words)

	PlanetMath: polytope
		is the convex hull of a finite set of points in
		This is version 13 of polytope, born on 2004-02-03, modified 2004-04-19.
		Object id is 5544, canonical name is Polytope.
	planetmath.org /encyclopedia/FVector.html (270 words)

	[No title] (Site not responding. Last check: 2007-10-12)
		For example, for d = 3: The convex hull of a set of points in 3-space (the convex hull is a 3-polytope) is simplicial iff every facet is a 2-simplex (a triangular convex hull of exactly 3 points).
		Recall that in a simplicial d-polytope, each facet is a (d-1)-simplex, and is determined by exactly d vertices.
		In a simplicial d-polytope, a subfacet is shared by exactly two facets, and two facets F1 and F2 share a subfacet e iff e is determined by a common subset, with d - 1 vertices, of the sets determining F1 and F2.
	longwood.cs.ucf.edu /courses/cot5520/Lec13.ppt (262 words)

	Polytechnic University Department of Mathematics: Klain (Site not responding. Last check: 2007-10-12)
		It is easy to see that a convex polygon in the plane is uniquely determined (up to translation) by the directions and lengths of its edges.
		First, the existence of a polytope satisfying given boundary data is demonstrated.
		In the second step, the uniqueness of that polytope (up to translation) is then shown to follow from the equality conditions of the Minkowski's inequality, a generalized isoperimetric inequality for mixed volumes that is typically proved in a separate context.
	www.math.poly.edu /news/notes/klain.phtml (217 words)

	Linear Programming - Simplex Method
		The simplex method starts at the origin and follows a path along the edges of the polytope to the vertex where the maximum occurs.
		In two dimensions a convex polytope is a region that is the intersection of a finite set of half-planes (the general idea of a
		In three dimensions a convex polytope is solid region that is the intersection of a finite set of half-spaces (the generalized idea of a
	math.fullerton.edu /mathews/n2003/LinearProgrammingMod.html (466 words)

	Four Dimensional Figures Page
		Also just for the record, a convex polytope is one with this property: If a line segment’s end points both lie in the polytope’s interior, then all the points on the line segment between the end points also lie in the polytope’s interior; and none of the faces overlap.
		Likewise, all the vertices of an n-dimensional uniform polytope are constrained by symmetry to lie on a single n-dimensional sphere centered at the polytope’s center of symmetry.
		The existence of exactly those 18 convex uniform polyhedra is a fundamental property of three-dimensional Euclidean space, and the existence of the 64 corresponding polychora is likewise a fundamental property of four-dimensional Euclidean space.
	hometown.aol.com /Polycell/uniform.html (4231 words)

	Prof. Jeff Erickson Abstract (Site not responding. Last check: 2007-10-12)
		I'll discuss two combinatorial questions about convex polytopes that come from the study of convex hull algorithms.
		First, say that a facet of a convex polytope is degenerate if it is not a simplex.
		Next, say that a polytope is fat if it has few vertices and few facets but many faces of intermediate dimension.
	www.math.uiuc.edu /Bulletin/Abstracts/October/oct16-98geompot.html (169 words)

	Computing Farthest Neighbors on a Convex Polytope (ResearchIndex) (Site not responding. Last check: 2007-10-12)
		Abstract: Let N be a set of n points in convex position in R 3.
		The farthest-point Voronoi diagram of N partitions R 3 into n convex cells.
		We consider the intersection G(N) of the diagram with the boundary of the convex hull of N.
	citeseer.ist.psu.edu /475112.html (353 words)

	dD Convex Hulls and Delaunay Triangulations (Site not responding. Last check: 2007-10-12)
		The convex hull class is parameterized by a traits class that provides
		The validity of the computed convex hull can be checked using the member funciton is_valid, which implements the algorithm of Mehlhorn et al.
		The class supports incremental construction of Delaunay triangulations and various kind of query operations (in particular, nearest and furthest neighbor queries and range queries with spheres and simplices).
	www.cgal.org /Manual/doc_html/cgal_manual/Convex_hull_d/Chapter_main.html (328 words)

	A Laplace transform algorithm for the volume of a convex polytope
		A Laplace transform algorithm for the volume of a convex polytope
		The computational complexity of both algorithms is essentially described by $n^m$, which makes them especially attractive for large $n$ and relatively small $m$, when the other methods with $O(m^n)$ complexity fail.
		The methodology which differs from previous existing methods uses a Laplace transform technique that is well suited to the half-space representation of $\Omega$.
	www.optimization-online.org /DB_HTML/2001/12/410.html (98 words)

	What is a dual of a convex polytope?
		What is a dual of a convex polytope?
		By the definition, a dual polytope has the same dimension as
		The duality theorem states that every convex polytope admits a dual.
	www.cs.mcgill.ca /~fukuda/soft/polyfaq/node7.html (68 words)

	What is convex polytope/polyhedron? (Site not responding. Last check: 2007-10-12)
		is called a convex polyhedron if it is the set of solutions to a finite system of linear inequalities, and called convex polytope if it is a convex polyhedron and bounded.
		When a convex polyhedron (or polytope) has dimension
		For the sequel, we might omit convex for convex polytopes and polyhedra, and call them simply polytopes and polyhedra.
	www.cs.mcgill.ca /~fukuda/soft/polyfaq/node4.html (58 words)

	Convex Polyhedron
		What is the face lattice of a convex polytope
		is in the convex hull of a given finite set
		Is there any efficient algorithm to compute the volume of a convex polytope in
	www.ifor.math.ethz.ch /~fukuda/polyfaq/node3.html (181 words)

	DC MetaData for: Dropping a Vertex or a Facet from a Convex Polytope (Site not responding. Last check: 2007-10-12)
		DC MetaData for: Dropping a Vertex or a Facet from a Convex Polytope
		Dropping a Vertex or a Facet from a Convex Polytope
		convex polytope in $\R^n$ having $N\geq c_0^n/\epsilon$ vertices
	www.esi.ac.at /Preprint-shadows/esi732.html (78 words)

	Extremal Properties for Dissections of Convex 3-Polytopes
		Extremal Properties for Dissections of Convex 3-Polytopes: SIAM Journal on Discrete Mathematics Vol.
		A dissection of a convex d-polytope is a partition of the polytope into d-simplices whose vertices are among the vertices of the polytope.
		Triangulations are dissections that have the additional property that the set of all its simplices forms a simplicial complex.
	epubs.siam.org /sam-bin/dbq/article/36623 (164 words)

	Approximating Shortest Paths on a Convex Polytope in Three Dimensions - Agarwal, Har-Peled, Sharir, Varadarajan ...
		0.9: Approximating Shortest Paths on a Convex Polytope..
		Agarwal, P.K., Har-Peled, S., Sharir, M., Varadarajan, K.R., Approximating shortest paths on a convex polytope in three dimensions, Journal of the ACM, 1997 (to appear).
		73 A linear algorithm for determining the separation of convex..
	citeseer.ist.psu.edu /316165.html (727 words)

	Strange Unfoldings of Convex Polytopes
		An unfolding of a convex polytope P in R^3 is a planar embedding of its boundary obtained by cutting the edges of some spanning tree T of the graph of P and flattening the boundary along the remaining edges.
		For the question (b), Tomomi Matsui (tomomi@misojiro.t.u-tokyo.ac.jp) constructed a polytope with 6 facets and 5 vertices which admits an ambiguous unfolding:
		Note that a question (related to (a')) on the existence of an unfolding without overlaps, where it is allowed to cut any place in the boundary, was answered positively by Aronov and O'Rouke [AO91].
	www.ifor.math.ethz.ch /~fukuda/unfold_home/unfold_open.html (492 words)

	Contents
		Is there an efficient way of determining whether a given point is in the convex hull of a given finite set of points in ?
		Is there any efficient algorithm to compute the intersection of two (or) polytopes
		Is there any efficient algorithm to compute the volume of a convex polytope in ?
	www.ifor.math.ethz.ch /~fukuda/polyfaq/node1.html (291 words)

	Citebase - A Note on the Size of the Largest Ball Inside a Convex Polytope (Site not responding. Last check: 2007-10-12)
		A Note on the Size of the Largest Ball Inside a Convex Polytope
		Use the Correlation Generator to explore the correlation between download impact ("hits") and citation impact.
		Citation coverage and analysis is incomplete and hit coverage and analysis is both incomplete and noisy.
	www.citebase.org /cgi-bin/citations?id=oai:arXiv.org:math/0505301 (263 words)

	Softwares
		I have developed a (recursive) program to compute the exact volume of a convex polytope in R_n, given by its defining hyperplanes Ax Lasserre J.B. "An analytical expression and an algorithm for the volume of a convex polyhedron in Rn", J. Optim.
		, a tool for the algorithmic treatment of convex polyhedra, developed in the Discrete Geometry group at the Institute of Mathematics of Technische Universitat, Berlin.
		This volume computation occurs in various applications in Economics, computational complexity analysis, linear systems modeling, VLSI design, Statistics....
	www.laas.fr /~lasserre/softwares.html (254 words)

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