I'm reading the proof of the cone theorem from Koll'ar-Mori and I find step 3 to be very unclear. In particular below is a simple proof of this step which must be wrong somehow. I am wondering where I went wrong.
The setup for this step is that we have the closed cone of curves $\overline{NE}$ and we define a closed convex(?) subcone $W = \overline{\overline{NE}_{K \geq 0} + \sum_{\dim F_L = 1} F_L}$ where we define $F_L = L^\perp \cap \overline{NE}$ where $L$ is a nef divisor, and $K = K_X + \Delta$ is the log canonical divisor of some log pair $(X, \Delta)$ (ie. $X$ is normal and $K_X + \Delta$ is $\mathbb{Q}$-Cartier). It is claimed in the book that $W = \overline{NE}$.
But isn't $W$ a closed convex subcone? Then, we know that it is equal to the convex hull of its extremal rays, so to show it is all of $\overline{NE}$, it would seem sufficient to show that it contains all of the extremal rays of $\overline{NE}$.
But then an extremal ray of $\overline{NE}$ is nothing but a one-dimensional face of the cone $\overline{NE}$, and faces are all of the form $L^\perp \cap \overline{NE}$ where $L \in \overline{NE}^\vee = \operatorname{Nef}$ is a nef divisor. But then, this means that all the extremal rays of $\overline{NE}$ are all of the form $F_L$, so they are all contained in $\overline{NE}_{K \geq 0} + \sum_{\dim F_L = 1} F_L$ before even passing to the closure, so $\overline{NE} \subset W$ as required.
What is wrong with this argument? Surely there has to be something I am misunderstanding as it avoids a complicated argument in the book. I suspect it's wrong somehow to assume that $W$ is convex but it seems clear that it is.
Thanks!
EDIT: It is also assumed in the book that $(X, \Delta)$ is a klt pair, but I don't think this assumption is relevant in this step. I may be wrong though, so I wanted to mention it here ration than including it in the main body.
I think the point of this step is that the $L$ over which the sum is taken are integral Cartier divisors. (Look at the statement of Theorem 3.15: the $\xi_i$ are supposed to be elements of $N_{\mathbb Z}$.) Your statement that "faces are all of the form.." is true, but it is not obvious a priori why for a given face in the $K \geq 0$ halfspace, we can choose the corresponding $L$ to be integral.