convex polytopes are often simply called "polytopes". Grünbaum points out that this is solely to avoid the endless repetition of the word "convex",...

following table lists some properties of the six convex regular 4-polytopes. The symmetry groups of these 4-polytopes are all Coxeter groups and given in the notation...

Convex Polytopes is a graduate-level mathematics textbook about convex polytopes, higher-dimensional generalizations of three-dimensional convex polyhedra...

cell, consists of a polyhedron. A polytope may be convex. The convex polytopes are the simplest kind of polytopes, and form the basis for several different...

related to the infinite 2D square tiling. Convex 4-polytopes can be cut and unfolded as nets in 3-space. A 4-polytope is a closed four-dimensional figure....

A.; Fukuda, K. (2000), "Exact Volume Computation for Polytopes: A Practical Study", Polytopes — Combinatorics and Computation, pp. 131–154, CiteSeerX 10...

treatment of the polytopes regularly derived from the regular polytopes, followed Boole-Stott's notations, enumerating the convex uniform polytopes by symmetry...

the convex hulls of indicator vectors of solutions to a combinatorial problem. If the facets of these polytopes can be found, describing the polytopes as...

4-dimensional cross-polytope also goes by the name hexadecachoron or 16-cell. It is one of the six convex regular 4-polytopes. These 4-polytopes were first described...

(of arbitrary orientation) is (the interior of) a convex polytope, or in the 2-dimensional case a convex polygon. An open interval is a connected open set...

regular polytopes in Euclidean, spherical and hyperbolic spaces. This table shows a summary of regular polytope counts by rank. Only counting polytopes of...

algorithmic treatment of convex polyhedra. Albeit primarily a tool to study the combinatorics and the geometry of convex polytopes and polyhedra, it is by...

of a face of a convex set, given below. In other areas of mathematics, such as the theories of abstract polytopes and star polytopes, the requirement...

specific polytopes (especially 0-1 polytopes, whose vertices are subsets of a hypercube) arising from integer programming problems. A face of a convex polytope...

describe a more general class of convex polytopes in three or more dimensions, obtained by gluing together polytopes combinatorially equivalent to hypercubes...

triangle is strictly convex. Convex curve – Type of plane curve Concave polygon – Simple polygon which is not convex Convex polytope – Convex hull of a finite...

Dimensions. Convex uniform polytopes: 1940: The search was expanded systematically by H.S.M. Coxeter in his publication Regular and Semi-Regular Polytopes. Nonregular...

forms a convex uniform honeycomb in which the cells are zonohedra. 1900: Thorold Gosset enumerated the list of semiregular convex polytopes with regular...

centers of the 122. These polytopes are from a family of 39 convex uniform polytopes in 6-dimensions, made of uniform polytope facets and vertex figures...

Ziegler, Günter M. (1995). "Chapter 4: Steinitz' Theorem for 3-Polytopes". Lectures on Polytopes. Graduate Texts in Mathematics. Vol. 152. Springer-Verlag...

a theorem in geometry, named after Augustin Cauchy. It states that convex polytopes in three dimensions with congruent corresponding faces must be congruent...

complexes of convex polytopes", Mathematische Annalen, 210: 7–16, doi:10.1007/BF01344542, MR 0350623 Matteo, Nicholas (2015), Convex Polytopes and Tilings...

points. Integral polytopes are also called lattice polytopes or Z-polytopes. The special cases of two- and three-dimensional integral polytopes may be called...

geometry to more than three dimensions and discovered all their convex regular polytopes, including the six that occur in four dimensions. The Schläfli...

5-polytope is contained in the 5-polytope or its interior; otherwise, it is non-convex. Self-intersecting 5-polytopes are also known as star polytopes,...

upper bound theorem states that cyclic polytopes have the largest possible number of faces among all convex polytopes with a given dimension and number of...

on the rigidity of convex polytopes. Special polytopes are also considered, such as permutohedra, associahedra and Birkhoff polytopes. Combinatorial geometry...

polyhedra, such as the Cauchy theorem on rigidity of convex polytopes. The study of regular polytopes, Archimedean solids, and kissing numbers is also a...

k-neighborly polytope is a convex polytope in which every set of k or fewer vertices forms a face. For instance, a 2-neighborly polytope is a polytope in which...

Algorithms that construct convex hulls of various objects have a broad range of applications in mathematics and computer science. In computational geometry...