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I wonder if there are any is at least one proof of any mathmatical fact that would use convex hull or convex envelope as mathmatical object valuable for that proof?

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Gauss-Lucas? – J. M. Dec 17 '11 at 16:40
Also Newton polygon, but the Wikpedia article does not do it justice. See also – lhf Dec 17 '11 at 18:43

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