# Extend space to make polyhedra convex hulls of finite sets

A (convex) polytope is the convex hull of a finite number of points in Euclidean space (this is the so-called "vertex description"). Alternatively, it can defined to be a bounded polyhedron (this is the "facet description").

Here a (convex) polyhedron is the intersection of a finite number of closed half-spaces.

My question is whether we can get a vertex description of polyhedra, and not just polytopes: namely, is there some reasonable way to topologically extend Euclidean space so that all polyhedra are convex hulls of finite sets of points in this larger space?

• Doesn't real projective space $\mathbb{RP}^n$ have precisely the property you desire? – Adam P. Goucher Oct 6 '17 at 15:35
• I was thinking something along those lines, but I’m not even sure convex hull is well-defined for projective space. – Sam Hopkins Oct 6 '17 at 15:48
• There's also the Minkowski-Weyl theorem, which says that every polyhedron is finitely generated (by a set of points and directions). cs.mcgill.ca/~fukuda/soft/polyfaq/node14.html – Tony Huynh Oct 6 '17 at 17:46
• @SamHopkins Oh, and I guess you don't want to identify antipodal points either -- so just $\mathbb{R}^n$ together with a 'sphere at infinity' should suffice. You'll need to define convex combinations of points on the sphere at infinity, though, which is difficult in the case of a pair of antipodal points. – Adam P. Goucher Oct 6 '17 at 17:49

I am not sure that I understand the question, but unbounded convex polyhedra in $R^3$ are determined, up to a translation, by their vertices and recession cone; see Section 1.4 in Alexandrov's book on Convex polyhedra. Roughly speaking, the recession cone encodes the behavior of the polyhedron at infinity. See also Theorem 3 on page 391 of that book, where it is discussed how to determine an unbounded convex polyhedron from the projection of its vertices into some plane, the curvatures at these vertices, and a given convex polyhedral cone.
By the way, the space of unbounded convex hypersurfaces of $R^n$ which are homeomorphic to $R^{n-1}$ and the recession cones of the corresponding convex bodies are extensively studied in