2013/10/15 by Andreas F. Holmsen, Holmsen, Andreas F., Edgardo Roldán-Pensado +1
Computer Science · Mathematics · #Computational Geometry and Mesh Generation #FOS: Mathematics #Mathematics and Applications #Metric Geometry (math.MG) #Point processes and geometric inequalities #math.MG
paper · pdf · doi:10.48550/arxiv.1310.4226
arxiv created 2013/10/15 · openalex publication_date 2013/10/15 · arxiv updated 2013/10/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Hadwiger's transversal theorem gives necessary and sufficient conditions for a family of convex sets in the plane to have a line transversal. A higher dimensional version was obtained by Goodman, Pollack and Wenger, and recently a colorful version appeared due to Arocha, Bracho and Montejano. We show that it is possible to combine both results to obtain a colored version of Hadwiger's theorem in higher dimensions. The proofs differ from the previous ones and use a variant of the Borsuk-Ulam theorem. To be precise, we prove the following. Let F be a family of convex sets in \mathbb Rd in bijection with a family P of points in \mathbb Rd-1. Assume that there is a coloring of F with sufficiently many colors such that any colorful Radon partition of points in P corresponds to a colorful Radon partition of sets in F. Then some monochromatic subfamily of F has a hyperplane transversal.