Titlebook: Convexity and Discrete Geometry Including Graph Theory; Mulhouse, France, Se Karim Adiprasito,Imre Bárány,Costin Vilcu Conference proceedin

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Reflections of Planar Convex BodiesIt is proved that every convex body in the plane has a point such that the union of the body and its image under reflection in the point is convex. If the body is not centrally symmetric, then it has, in fact, three affinely independent points with this property.

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Envelopes of ,-SectionsLet . be a planar convex body af area |.|, and take .. An .-section of . is a line cutting . into two parts, one of which has area .. This article presents a systematic study of the envelope of .-sections and its dependence on .. Several open questions are asked, one of them in relation to a problem of fair partitioning.

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A Science of Mathematical Education,g result: For any planar convex bodies .,?. with ., and any ., there exists an .-section of . which is a .-section of . for some .. (An .-section of . is a straight line cutting . into two parts, one of which has area .). The question remains open if the word “planar” is dropped.

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