polytope

A polytope is the convex hull of finitely many points inEuclidean space. A polytope constructed in this way is the convexhull of its vertices and is called a $\mathrm{V}$-polytope.An $\mathrm{H}$-polytope is a bounded intersection ofupper halfspaces. By the Weyl–Minkowski theorem, these descriptions areequivalent, that is, every $𝒱$-polytope is an$\mathscr{H}$-polytope, and vice versa. This shows that ourintuition, based on the study of low-dimensional polytopes, that onecan describe a polytope either by its vertices or by its facets isessentially correct.

The dimension of $P$ is the smallest $d$ such that $P$ can beembedded in ${ℝ}^d$. A $d$-dimensional polytope is also calleda $d$-polytope.

A face of a polytope is the intersection of the polytope with asupporting hyperplane. Intuitively, a supporting hyperplane is ahyperplane that “just touches” the polytope, as though the polytopewere just about to pass through the hyperplane. Note that thisintuitive picture does not cover the case of the empty face, where thesupporting hyperplane does not touch the polytope at all, or the factthat a polytope is a face of itself. The faces of a polytope, whenpartially ordered by set inclusion, form a geometric lattice, calledthe face lattice of the polytope.

The Euler polyhedron formula, which states that if a $3$-polytope has$V$ vertices, $E$ edges, and $F$ faces, then

$$V-E+F=2,$$

has a generalization to all $d$-polytopes. Let $(f_{-1}=1,f_0,\mathrm{\dots },f_{d-1},f_d=1)$ be the f-vector of a $d$-polytope $P$, so$f_i$ is the number of $i$-dimensional faces of $P$. Then these numberssatsify the Euler–Poincaré–Schläfli formula:

$$\sum _{i=-1}^d{(-1)}^i{f}_i=0.$$(1)

This is the first of many relations among entries of the f-vectorsatisfied by all polytopes. These relations are called theDehn–Sommerville relations. Any poset which satisfies theserelations is Eulerian (http://planetmath.org/EulerianPoset),so the face lattice of any polytope is Eulerian.