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0633 subsets in disjoint convex sets 6th edition Round 3 p3

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May 3, 2021
geometrycombinatorics

Problem Statement

We say that a set of points MM in the plane is convex if for any two points A,B∈MA, B \in M, all the points from the segment (AB)(AB) also belong to MM. Let n≥2n \ge 2 be an integer and let FF be a family of nn disjoint convex sets in the plane. Prove that there exists a line ℓ\ell in the plane, a set M∈FM \in F and a subset S⊂FS \subset F with at least ⌈n12⌉\lceil \frac{n}{12} \rceil elements such that MM is contained in one closed half-plane determined by ℓ\ell, and all the sets N∈SN \in S are contained in the complementary closed half-plane determined by ℓ\ell.
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