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Sep 1, 2021 at 17:47 comment added Yemon Choi @Hugh Your original question had a ttle which asked for "understanding" of the intersection number, and so my answer was meant to suggest that it can be viewed geometrically; indeed, the measure $\mu$ that is provided by the theorem you quote comes from a separating hyperplane argument IIRC. You even say "I am having a hard time understanding what is this intersection number concretely" and so I tried to offer a geometric formulation. If all you wanted was a walk-through of a simple solution to the exercise/example that you provided, then perhaps you could reword your question to that effect.
Sep 1, 2021 at 17:46 comment added Yemon Choi @Hugh if you are referring to my proof that a particular set system has intersection 2/5, then I was not aiming to find the shortest proof, but rather to show how the geometric perspective with convex hulls could be applied.
Sep 1, 2021 at 8:00 comment added Hugh Thank you for having taken the trouble to answer, but I have a feeling that your argument is overcomplicated. There must be a simpler approach.
Aug 31, 2021 at 19:39 history edited Yemon Choi CC BY-SA 4.0
Added a worked example which should correspond to the example mentioned in the OP's question.
Aug 31, 2021 at 19:29 history answered Yemon Choi CC BY-SA 4.0