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I'm writing on behalf of a group project where we are currently looking at basic geometry; in particular we are interested in polyhedral fans. We wish to prove that (abusing terminology somewhat) the intersection of two fans is a fan, and our primary stumbling block at this point is the following lemma:

Suppose that $\sigma$ and $\tau$ are cones (these have vertex at the origin), and $f$ is a face of $\sigma\cap\tau$. Prove that there are faces $s\leq\sigma$ and $t\leq\tau$ such that $f=s\cap t$.

Our idea is to choose $s$ and $t$ to be the minimal faces of $\sigma$ and $\tau$ which contain $f$. But we are having a hard time dealing with what "minimal" means. We can define it (just the minimum element in the sublattice of faces that contain $f$), but have not been able to translate the definition into a useful property that shows $s\cap t \subseteq f$.

Eric Stucky
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Cones are defined by linear inequalities, and for a fan you have that all half spaces defined by an inequality pass through the origin, and when some of these inequalities are saturated (forming linear equations) this defines a face of the cone. So if you intersect two cones, you are simply combining their lists of inequalities and thus a face of the intersection arises when some of those inequalities become equations. By assigning each equation to the appropriate cone(s) which have an inequality leading to that equation, you get two faces (one face of each cone) whose intersection gives you the face of the intersection cone. Note that the "face" of one cone so obtained might actually be the full-dimensional cone, e.g. if a face of one cone cuts through the interior of the other cone.

user2566092
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  • Haha okay, that is very simple. Unfortunately we are using a different definition of "cone" so I will see if I can prove it is equivalent to yours; that might be easier and actually it is probably in the literature. – Eric Stucky Jul 14 '15 at 19:36
  • @EricStucky When you said "fan" I automatically thought of normal fans of polyhedra whose cells are well-known to be defined by inequalities with RHS = 0. If your definition of fan turns out that it can't be described as the complex obtained by an arrangement of hyperplanes passing through the origin, I'll delete my answer. – user2566092 Jul 14 '15 at 19:38
  • I'm relatively confident it can: our cones are just positive linear combinations of a finite set of vectors. In 1,2,3-d these things are pretty obviously equivalent; my intuition says they are the same all the way up. – Eric Stucky Jul 14 '15 at 19:40