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Journal Articles Periodica Mathematica Hungarica Year : 2023

Covering families of triangles


A cover for a family F of sets in the plane is a set into which every set in F can be isometrically moved. We are interested in the convex cover of smallest area for a given family of triangles. Park and Cheong conjectured that any family of triangles of bounded diameter has a smallest convex cover that is itself a triangle. The conjecture is equivalent to the claim that for every convex set X there is a triangle Z whose area is not larger than the area of X , such that Z covers the family of triangles contained in X. We prove this claim for the case where a diameter of X lies on its boundary. We also give a complete characterization of the smallest convex cover for the family of triangles contained in a half-disk, and for the family of triangles contained in a square. In both cases, this cover is a triangle.
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hal-03662311 , version 1 (09-05-2022)





Otfried Cheong, Olivier Devillers, Ji-Won Park, Marc Glisse. Covering families of triangles. Periodica Mathematica Hungarica, 2023, 87, pp.86--109. ⟨10.1007/s10998-022-00503-4⟩. ⟨hal-03662311⟩
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