Discrete & Computational Geometry

, Volume 17, Issue 3, pp 307–318

# A combinatorial property of convex sets

• M. Abellanas
• G. Hernandez
• R. Klein
• V. Neumann-Lara
• J. Urrutia
Article

## Abstract

A known result in combinatorial geometry states that any collection Pn of points on the plane contains two such that any circle containing them contains n/c elements of Pn, c a constant. We prove: Let Φ be a family of n noncrossing compact convex sets on the plane, and let S be a strictly convex compact set. Then there are two elements Si, Sj of Φ such that any set S′ homothetic to S that contains them contains n/c elements of Φ, c a constant (S is homothetic to S if 5’ = λS + v, where λ is a real number greater than 0 and v is a vector of ℜ2). Our proof method is based on a new type of Voronoi diagram, called the “closest covered set diagram” based on a convex distance function. We also prove that our result does not generalize to higher dimensions; we construct a set Φ of n disjoint convex sets in ℜ3)3 such that for any nonempty subset ΦHh of Φ there is a sphere SH containing all the elements of ΦH, and no other element of Φ.

## Keywords

Nonempty Subset Voronoi Diagram Convex Polygon Combinatorial Property Voronoi Region
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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