Locally polyhedral and locally finitely generated cones

Question

Let $V$ be a finite dimensional real vector space. The following definitions are taken from a paper.

A closed convex cone $K \subset V$ is locally polyhedral at $v \in \partial K$ if there exists a neighbourhood $U=U(v)$ in V such that $K$ is defined by a finite number of inequalities in $U$. The cone $K$ is said locally polyhedral if it is locally polyhedral at any point $v \in \partial K$

We say that $K$ is locally finitely generated at $v$ if there exist a closed subcone $C \subset K$ and a finite set of vectors $\mathcal{V}=\{v_1,\dots,v_k\}$ such that $v \notin C$ and $K$ is generated by $C$ and $\mathcal{V}$.

I have a couple of questions:

1. If $K$ is locally polyhedral at any point, actually shouldn't it be polyhedral?
2. Isn't it true that "K is locally polyhedral at $v$ if and only if it is locally finitely generated at $v$?

I cannot imagine patologies where locally finitely generated at $v$ doesn't imply locally polyhedral at $v$, or where "locally polyhedral" (everywhere) doesn't imply "polyhedral" (at least if the cone doesn't contain lines). Can you provide some if you have?

Thanks a lot.

Answer 1

Both questions have affirmative answers. See the paper of A. Bastiani, Polyèdres convexes dans les espaces vectoriels topologiques, 1959 Séminaire C. Ehresmann, 1957/58, no. 19, 46 pp. Faculté des Sciences de Paris.