Is there a term for an "unbounded simplex"?

Question

Is there a general term for regions like $\{(x,y):x>y\}$ and $\{(x,y,z): x>y>z\}$, i.e., regions which are simplexes with one open?

Answer 1

Maybe you're looking for an unbounded polytope?

Answer 2

I was somehow brought up with the term "polytopal cone", but it doesn't seem to be standard at all.

I dislike "polyhedra" because to me, those are just $3$-dimensional polytopes.