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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?

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  • $\begingroup$ Doesn't real projective space $\mathbb{RP}^n$ have precisely the property you desire? $\endgroup$ – Adam P. Goucher Oct 6 '17 at 15:35
  • $\begingroup$ I was thinking something along those lines, but I’m not even sure convex hull is well-defined for projective space. $\endgroup$ – Sam Hopkins Oct 6 '17 at 15:48
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    $\begingroup$ 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 $\endgroup$ – Tony Huynh Oct 6 '17 at 17:46
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    $\begingroup$ @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. $\endgroup$ – Adam P. Goucher Oct 6 '17 at 17:49
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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.

The cone, in turn, may be determined by a convex polygon in the plane mentioned above and the apex of the cone. In short there is a way to parametrize the space of unbounded convex polyhedra in terms of a finite number of points, which prescribe its vertices and its recession cone. In this conception, the unbounded polyhedron is just the convex hull of the union of its vertices together with the recession cone, once it is properly positioned relative to the vertices.

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

Deformations of unbounded convex bodies and hypersurfaces, Amer. J. Math.,134 (2012),1585-1611

where it is shown for instance that the space of these convex hypesurfaces admits a strong deformation retraction into the Grassmannian space of hyperplanes. This is true in the polyhedral category as well.

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