3. Convex polyhedra

By a polyhedron let us mean a solid in R$^3$ whose boundary consists of finitely many planar polygons. Examples are a cube and the ``regular polyhedra'' shown in Figures , , and .

This definition is too informal, however. We don't want to allow a solid that is in two or more pieces; we don't want to allow a solid that extends infinitely far (such as the part of R$^3$ outside a tetrahedron); we don't want to allow a solid that doesn't include its boundary (such as the ``open'' cube described by $0 < x < 1$, $0 < y < 1$, $0 < z < 1$); and we don't want to allow a solid with no thickness (for example, a single triangle).

A better definition is to say that a polyhedron is a solid that can be obtained by gluing together finitely many tetrahedra. A cube could be made in this way, for example.

Note that the plural of ``polyhedron'' is ``polyhedra''¹.

The easiest kind of polyhedron to deal with is a convex polyhedron.

Proposition 4.1. A subset $C$ of R$^3$ is a convex polyhedron if and only if $C$ is the convex hull of a finite set and is not contained in a plane.