# Name for facet of a cone containing all but one edge

Let $$C \subseteq \mathbb R^n$$ be a polyhedral cone, so generated by its edges ($$1$$-dimensional faces) and $$F \subseteq C$$ a facet (codimension $$1$$ face) of it containing every edge except $$e$$. In particular, the map $$F \times e \stackrel{+}{\to} C$$ is an isomorphism of cones.

Is there a name for this property of $$F$$ or $$e$$ (which determine one another)?

Essentially, I regard the edges $$e$$ like this as kind of trivial, and would like to split them off to deal with the difficult part of $$C$$. So maybe one could speak of a "core facet" $$F$$ of $$C$$, and let the "core of $$C$$" be the intersection of the "core facets". Then the map $$core(C) \times \prod_{\text{core edges }e} e \stackrel{+}{\to} C$$ would be an isomorphism, and $$core(C)$$ would have no core edges.

There is a similar, familiar construction in the theory of simplicial complexes, where a "cone vertex" $$v$$ of $$\Delta$$ is one lying in every maximal face. One can safely delete all the cone vertices, and recone on them to reconstruct $$\Delta$$. Obviously one doesn't want to steal this terminology directly and speak of "cone edges".

EDIT: If you prefer, you can slice the (pointed) cone with a hyperplane to get a polytope, and work with that. It's then common to say "this polytope $$P$$ is the cone on this facet $$F$$, from this vertex $$e$$" but I still don't know a good adjective for $$F$$ or $$e$$.

• Sorry I don't understand. Every facet of a (general position) simplicial cone in $\mathbb R^3$ contains all edges but one, no? – მამუკა ჯიბლაძე Dec 31 '18 at 12:46
• That's correct; I'm not asking about simplicial cones. Think instead of the cone on a square, i.e. a pyramid with no base. – Allen Knutson Dec 31 '18 at 13:58
• You mean a cone over this pyramid? In 4d? – მამუკა ჯიბლაძე Dec 31 '18 at 15:10
• I'd meant a cone on a square, in 3d. That cone has no edges of the special type I'm asking about. But if you want we can consider the 4d cone on a pyramid (with its base). That cone has 5 edges, of which only the line through the apex is special in the sense I'm asking about. – Allen Knutson Dec 31 '18 at 18:17
• Maybe you prefer the statement "$Proj(R\otimes S)$ is the join of $Proj(R)$ and $Proj(S)$", where $R$ and $S$ are the monoid algebras, and "join" means the union of the lines through the two spaces? – Allen Knutson Dec 31 '18 at 18:55