Convex cone generated by extreme rays

Question

Let $X$ be a vector space and $K \subseteq X$ be a pointed convex cone. Let $L$ denote the set of extreme rays of $K.$ The questions are: under which condition can I guarantee that $$K= cone(conv(L))?$$ Here, $cone(A)=\{\lambda x: x\in A, \; \lambda \geq 0\}$ and $conv(A)$ is the convex hull of $A.$ Any reference that treats this problem? I am particularly interested in the infinite dimensional case. Thanks in advance

Answer 1

This appears to be handled in "Convex Analysis" By Rockafellar, Theorem 18.5 ( on page 166).

THEOREM 18.5. Let C be a closed convex set containing no lines, and let S be the set of all extreme points and extreme directions of C. Then C = conv(S).