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Radon's lemma states:

Let A be a set of $d$+2 points in $R^{d}$. Then there exist two disjoint subsets $A1$, $A2$ $\subset$ $A$ such that

$conv(A1)$ $\cap$ $conv(A2)$ = $\emptyset$

I understand that this is true and I am not looking for a proof, however I am not sure why it is useful. I have seen that it is used in Helly's Theorem, however I do not understand how showing that there could exist a non-empty intersection if we take the convex hulls of two sets says anything about the actual existence of intersections. I think I am missing some fundamental understanding of what this lemma actually means.

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