Convex cone generated by extreme rays 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
 A: 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).
A: Nothing of this sort is true for most standard cones (e.g. the natural cone in classical function spaces). Consider the following examples.

Example 1. Let $X = C_{\mathbb{R}}[0,1]$ with its usual cone. Then an extreme ray corresponds with a function $f \in X_+ \setminus \{0\}$ such that $0 \leq g \leq f$ implies $g = \alpha f$. But any non-zero function must be non-zero on some open interval, so for any $f \gneq 0$ there is a plethora of functions lying between $0$ and $f$. Conclusion: there are no extreme rays.


Example 2. Let $Y = \ell_{\mathbb{R}}^\infty$ with its usual cone. The extreme rays are the standard basis vectors $e_i$, so the closed convex cone they generate is only $(c_0)_+$.


Example 3. Similarly, let $Z = B(\ell_{\mathbb{C}}^2)^{\text{sa}}$ (the self-adjoint operators $\ell_{\mathbb{C}}^2 \to \ell_{\mathbb{C}}^2$) with the positive semidefinite cone. The extreme rays are the rank one orthogonal projections. The closed cone they generate is $K(\ell_{\mathbb{C}}^2)_+$, the cone of compact positive semidefinite operators.

Similarly, the statement fails for many spaces of differentiable or Lebesgue integrable functions on some domain $U \subseteq \mathbb{R}^n$. (It is however true for most sequence spaces, for instance $\ell_{\mathbb{R}}^p$ with $1 \leq p < \infty$.)
To get sufficient criteria for the statement to be true, I guess one must resort to Krein–Milman type theorems (e.g. assume that the cone has a weakly compact base). For more on this, see §3.8 and §3.12 in [Jam70]. (Warning before reading Jameson's book: in the ordered vector spaces community, cone means proper/pointed convex cone — see §1.1.) In particular:

Proposition. Let $E$ be a locally convex space, and let $E_+ \subseteq E$ be a convex cone. If $E$ has an interior point, then the (topological) dual cone $E_+' \subseteq E'$ is the weak-$*$ closed convex cone generated by its extreme rays.
Proof. Combine Theorem 3.8.6, Theorem 3.12.8 and Corollary 3.12.9 from [Jam70]. $\hspace{18mm}\Box$

References.
[Jam70]: Graham Jameson, Ordered Linear Spaces, Springer Lecture Notes in Mathematics 141, 1970.
