Covering families of triangles

Otfried Cheong (1) , Olivier Devillers (2) , Ji-Won Park (2) , Marc Glisse (3)

1 CS - Department of Computer Science [KAIST]
2 GAMBLE - Geometric Algorithms and Models Beyond the Linear and Euclidean realm
3 DATASHAPE - Understanding the Shape of Data

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.

Domaines: Géométrie algorithmique [cs.CG]

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⟩