If I have a closed polygonal region in the plane, the tangent cone to a point of the polygon is either:
- The whole tangent space, at interior points;
- A half-plane, at interior points of edges;
- A proper convex cone, at convex vertices;
- _______, at non-convex vertices.
Is there a term for the last class of linear cones? I can think of a few ways to characterize it:
- They are precisely the closures of complements of proper convex cones.
- They are precisely the closed, connected cones containing a half-plane.
- They are precisely the unions of two distinct half-planes.
- They, together with the half-planes, are the cones which are the unions of the half-planes they contain.
Bonus points if there's a characterization that generalizes in some way to higher-dimensional polytopes.
Update
To address some issues raised in the comments:
- By a "polygonal region" I mean a compact subset of the plane whose boundary is the disjoint union of finitely many simple polygons.
- By a "simple polygon" I mean a finite set of distinct points equipped with a cyclic order, together with all the line segments between consecutive points, subject to the condition that no two non-consecutive edges intersect.
- By a "proper convex cone" in two dimensions I just mean it's two dimensional but not the entire space. (Obviously this leaves some question of how to generalize to higher dimensions.) I guess there's a question of whether to include the edge case of a half space, but at least in the two dimensional case it's easy to slightly reformulate things to handle either condition.
- I'm looking for an if and only if condition like the others. Arbitrary nonconvex cones can be things like unions of finitely many rays.