# 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”.