A good description of the set of inner points of a polyhedron.

Question

How can I get a good description of the inner points of a polyhedron? I am trying to calculate the volume of a polyhedron by change of variables, but I can't describe the set of points of the polyhedron properly (given its vertices). I look for a description such as {(x,y,z):x+y+z<3;x<4y}.

Answer 1

For a convex polyhedron this is not so hard: any convex polyhedron is the intersection of half-spaces defined by the faces (each face lies on exactly one plane, and the polyhedron lies entirely on one side of that plane). Thus, the set of points in a convex polyhedron can be specified by $F$ linear inequalities, where $F$ is the number of faces.

For a non-convex polyhedron, I think the easiest thing to do is to express the polyhedron as a union of convex polyhedra whose pairwise intersections all have measure $0$, then apply the same procedure as above. Then you do the change of variables to each convex piece and add up your answers.