**2-neighbourly**each pair of vertices defines an edge.

**References:**146**3-polytope**The convex hull of a finite set of points in

*R*^{3}.**4-polytope**The convex hull of a finite set of points in

*R*^{4}**algorithms****application****beneath**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**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**book****centered**A polytope is centered if the origin is contained in the interior of \conv

*V*\*v*for all*v**in**V***centrally symmetric**A polytope

*P*is called centrally symmetric if*x**in**P*implies -*x**in**P*.**References:**169**combinatorial structure****References:**153, 143, 145, 152, 196, 147, 140, 144, 141, 10, 139, 11, 146**computation****References:**37**computational geometry****containment****convex hull****convexity****References:**142**cyclic****data structures****References:**41**degeneracy****diagonal****diagonals****References:**194**diameter***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.**dissection**A

*dissection*of a*d*-polytope*P*is a set of*d*-polytopes {*Q*_{1},*Q*_{2}, ...,*Q*_{n}} such that*P*= \union*Q*_{i}and the interiors of the*Q*_{i}s are pairwise disjoint.**divide-and-conquer****References:**51**doctoral thesis****References:**51**domination****References:**51**double-description****dwarfed**A polytope

*P*is called dwarfed if (*P*) = (*Q*) \union*h*^{+}where*f*_{0}(*P*) <<*f*_{0}(*Q*)**References:**3**enumeration****equidecomposable**A polytope is called equidecomposable if every triangulation has the same

*f*-vector**References:**115**estimation****extreme-point computation****f-vector****face-enumeration****faces>>size***sum**f*_{k}>>*d*(*f*_{0}+*f*_{d-1})**facet-degenerate**(some) facets contain more than

*d*vertices, i.e. not simplicial.**facet-enumeration****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**facets>>vertices***f*_{d-1}>>*f*_{0}.**family description****References:**98**fixed-dimension****Fourier-Motzkin elimination****g-vector****general-position****References:**18**gift-wrapping****halfspace**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)**hyperplane**A

*hyperplane*is a set of points satisfying some linear inequality*ax*=*b*.**illumination****implementation****incidence graph****incremental***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**intersection****References:**51**k-sets****References:**26**linear inequality****References:**30**linear programming****linear-algebra****References:**142**lower bound****References:**11**Lower Bound Theorem****lower bounds****lower-bounds****References:**27**metric****Minkowski-Weyl Theorem****References:**30**neighbourly**each

*k*<*floor*(*d*/2) vertices forms a face.**non-convex****References:**9**NP-Complete****oriented matroid****References:**185**oriented-matroid****References:**118**output-sensitive****p-vector****parallel****parallelapiped****References:**9**perturbation****pivoting****References:**56, 58, 70, 77, 20, 14, 21, 55, 71, 6, 72, 28, 13, 62, 30, 27, 7, 31**plane-sweep****References:**51**polyhedra****References:**50**polytope****References:**37**polytope pairs****References:**171**pseudomanifold****References:**172**pulling****References:**122**pushing****References:**130**random****random sampling****References:**26**realization space****References:**185**redundancy-removal****References:**181**reference****ridge-diameter**The diameter (in the graph theoretic sense) of the dual polytope.

**rigidity****simple**Exactly

*d*facets intersect at each vertex.**simplicial**Exactly

*d*vertices on each facet.**skeleton**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.**stacked polytope**An

*n*-vertex*stacked*is either a*d*-polytope*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.**symmetry****References:**144**triangle-free***P*has no triangular 2-face.**References:**145**triangulation**A dissection of a polytope into simplices such that any pair intersect in a (possibly empty) face.

**truncation**Intersection with a halfspace that cuts off exactly one vertex.

**truncation polytope**An

*m*-facet*truncation*is either a simplex or the truncation of an (*d*-polytope*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.**References:**140**Upper Bound Theorem****References:**182**vertex**A 0-dimensional face of a polytope. Equivalently an extreme point.

**vertex-degenerate**(some) vertices are contained in more than

*d*facets, i.e. not simple. See also facet-degenerate.**vertex-enumeration****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**vertices>>facets**See facets>>vertices

**volume****References:**37**Voronoi diagram****weakly neighbourly****References:**115**zero-one**A polytope is called zero-one if every vertex coordinate has one exactly two values (e.g. 0 or 1).

**zonotope**A zonotope is the minkowski sum of a set of vectors.