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Dubovickii-Milyutin theorem

I am trying to understand the eponymous result in question and it seems that there is a dearth of treatments of it.

It says that if $K_{i},i=1,\ldots,n+1$ are convex cones in a Banach space, with $K_{i},i=1,\ldots,n$ open, then the intersection of all the cones is nonempty if and only if there are $x_{i} \in K_{i}^{*}$, not all zero, so that $\sum{x_{i}}=0$.

1. What is an accessible reference that treats this result in context? I've seen mentions of books by Girsanov and Zeidler but I don't have them available. Anywhere else, perhaps? A general reference on such questions (intersections of cones etc.) will also be greatly appreciated.

2. What if the set of cones is infinite? What is known for that case?

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