each pair of vertices defines an edge.
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The convex hull of a finite set of points in R3.
The convex hull of a finite set of points in R4
A point p is beneath a facet F of a polytope P if p is in the closed halfspace induced by F containing P. See also beyond
A point p is beyond a facet F of a polytope P if p is in the open halfspace induced by F not containing P. See also beneath
A polytope is centered if the origin is contained in the interior of \conv V\v for all v in V
A polytope P is called centrally symmetric if x in P implies -x in P.
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Diameter is used in two different senses. In the metric sense, it means the length of the longest line segment between points in a set (see e.g. [26]). In the combinatorial sense, it is used to mean the length of the longest path in the skeleton of a polytope, or of a graph in general. See also ridge-diameter.
A dissection of a d-polytope P is a set of d-polytopes { Q1, Q2, ..., Qn } such that P = \union Qi and the interiors of the Qis are pairwise disjoint.
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A polytope P is called dwarfed if (P) = (Q) \union h+ where f0(P) << f0(Q)
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A polytope is called equidecomposable if every triangulation has the same f-vector
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sum fk >> d(f0+fd-1)
(some) facets contain more than d vertices, i.e. not simplicial.
References: 22, 49, 100, 18, 8, 59, 61, 20, 60, 53, 63, 55, 2, 29, 10, 12, 13, 68, 62, 51, 11, 69, 5, 3
fd-1 >> f0.
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A halfspace is the set of points satisfying some linear inequality ax < b (i.e. an open halfspace) or ax < = b (i.e. a closed halfspace)
A hyperplane is a set of points satisfying some linear inequality ax = b.
Incremental algorithms for e.g. facet-enumeration proceed by adding the input points one by one, updating the list of facet-defining inequalities for the current intermediate polytope at each step. See also double-description and Fourier-Motzkin elimination
References: 22, 49, 61, 60, 35, 24, 63, 29, 10, 12, 13, 15, 62, 23, 27, 31
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each k < floor(d/2) vertices forms a face.
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References: 56, 58, 70, 77, 20, 14, 21, 55, 71, 6, 72, 28, 13, 62, 30, 27, 7, 31
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The diameter (in the graph theoretic sense) of the dual polytope.
Exactly d facets intersect at each vertex.
Exactly d vertices on each facet.
The skeleton of a convex polytope is the graph formed by its vertices and edges. The famous theorem of Steinitz [161, 191] says that the skeletons of 3-polyhedra are exactly the 3-connected planar graphs.
An n-vertex stacked d-polytope is either a d-simplex, or the convex hull of an (n-1)-vertex stacked polytope with an additional point that is beyond exactly one facet. These polytopes have the minimum number of facets for an n-vertex simplicial polytope. See also truncation.
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P has no triangular 2-face.
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A dissection of a polytope into simplices such that any pair intersect in a (possibly empty) face.
Intersection with a halfspace that cuts off exactly one vertex.
An m-facet truncation d-polytope is either a simplex or the truncation of an (m-1)-facet truncation polytope. These polytopes have the minimum number of vertices possible for an m-facet simple polytope. They are dual to the stacked polytopes.
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A 0-dimensional face of a polytope. Equivalently an extreme point.
(some) vertices are contained in more than d facets, i.e. not simple. See also facet-degenerate.
References: 16, 56, 58, 52, 1, 14, 60, 21, 35, 181, 24, 63, 34, 64, 71, 29, 6, 10, 12, 28, 68, 15, 23, 51, 27, 7, 57, 11, 69, 5, 3
See facets>>vertices
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A polytope is called zero-one if every vertex coordinate has one exactly two values (e.g. 0 or 1).
A zonotope is the minkowski sum of a set of vectors.