Given a polyhedral cone, its intersection with any two-dimensional plane is either a polygon or a region enclosed by a polygonal curve. Is it a characterization of polyhedral cones? Does there exists a convex cone, which is not polyhedral, but such that all its two-dimensional sections are regions enclosed by polygonal curves?
Victor Klee, "Some characterizations of convex polyhedra." Acta Mathematica 1959, Volume 102, Issue 1-2, pp 79-107.
I haven't read the whole paper, but Klee proved that if all $j$-dimensional sections of an $n$-dimensional convex region are polyhedral for $2\le j \le n-1$, then the region is polyhedral. His definition for polyhedral includes polyhedral cones and other unbounded polyhedra. For $j=2$ and a closed convex cone, Klee cites Mirkil for a slightly earlier proof.