## By keyword

centered

A polytope is centered if the origin is contained in the interior of \conv V\v for all v in V

centrally-symmetric

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. ). 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.

dwarfed

A polytope P is called dwarfed if (P) = (Q) \union h+ where f0(P) << f0(Q)

equidecomposable

A polytope is called equidecomposable if every triangulation has the same f-vector

faces>>size

sum fk >> d(f0+fd-1)

facet-degenerate

(some) facets contain more than d vertices, i.e. not simplicial.

facets>>vertices

fd-1 >> f0.

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

neighbourly

each k < floor(d/2) vertices forms a face.

ridge-diameter

The diameter (in the graph theoretic sense) of the dual polytope.

simple

Exactly d facets intersect at each vertex.

simplicial

Exactly d vertices on each facet.

triangle-free

P has no triangular 2-face.

triangulation

A dissection of a polytope into simplices such that any pair intersect in a (possibly empty) face.

truncationpolytope

vertex-degenerate

(some) vertices are contained in more than d facets, i.e. not simple. See also facet-degenerate.

vertices>>facets

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.

## By Name

centergon(n)
centered, simple, simplicial
cube(d)
vertices>>facets, simple, triangle-free, facet-degenerate, centrally-symmetric, triangulation, zero-one, zonotope
cut(n)
facet-degenerate
cyclic(n,d)
facets>>vertices, simplicial, neighbourly
dwarfcube(d)
simple, truncationpolytope, dwarfed, facet-degenerate
hamming(n)
hypersimplex(d,k)
zero-one
interval(a,b)
metric(n)
facet-degenerate, ridge-diameter
permutahedron(n)
simple, zonotope
piercecube(d)
incremental, facet-degenerate
prodcyclic(d,n)
incremental, vertex-degenerate, facet-degenerate, triangulation, faces>>size
prodsimplex(d)
simple, facet-degenerate, triangulation, equidecomposable
prodsumcube(d)
faces>>size, vertex-degenerate, facet-degenerate, triangulation, incremental, zero-one
prodsumpolygon(d,n)
incremental, vertex-degenerate, facet-degenerate, triangulation, centered, faces>>size
q4
simple, diameter
simplex(d)
simple, simplicial, truncationpolytope
sumcube(d)
centered, facets>>vertices, zero-one
sumpolygon(d,n)
facets>>vertices, simple, centered