A polytope is centered if the origin is contained in the interior of \conv V\v for all v in V
centergon(n) , prodsumpolygon(d,n) , sumcube(d) , sumpolygon(d,n)
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.
A polytope P is called dwarfed if (P) = (Q) \union h+ where f0(P) << f0(Q)
A polytope is called equidecomposable if every triangulation has the same f-vector
sum fk >> d(f0+fd-1)
prodcyclic(d,n) , prodsumcube(d) , prodsumpolygon(d,n)
(some) facets contain more than d vertices, i.e. not simplicial.
cube(d) , cut(n) , dwarfcube(d) , metric(n) , piercecube(d) , prodcyclic(d,n) , prodsimplex(d) , prodsumcube(d) , prodsumpolygon(d,n)
fd-1 >> f0.
cyclic(n,d) , sumcube(d) , sumpolygon(d,n)
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
piercecube(d) , prodcyclic(d,n) , prodsumcube(d) , prodsumpolygon(d,n)
each k < floor(d/2) vertices forms a face.
The diameter (in the graph theoretic sense) of the dual polytope.
Exactly d facets intersect at each vertex.
centergon(n) , cube(d) , dwarfcube(d) , permutahedron(n) , prodsimplex(d) , q4 , simplex(d) , sumpolygon(d,n)
Exactly d vertices on each facet.
centergon(n) , cyclic(n,d) , simplex(d)
P has no triangular 2-face.
A dissection of a polytope into simplices such that any pair intersect in a (possibly empty) face.
cube(d) , prodcyclic(d,n) , prodsimplex(d) , prodsumcube(d) , prodsumpolygon(d,n)
(some) vertices are contained in more than d facets, i.e. not simple. See also facet-degenerate.
prodcyclic(d,n) , prodsumcube(d) , prodsumpolygon(d,n)
See facets>>vertices
A polytope is called zero-one if every vertex coordinate has one exactly two values (e.g. 0 or 1).
cube(d) , hypersimplex(d,k) , prodsumcube(d) , sumcube(d)
A zonotope is the minkowski sum of a set of vectors.