The colored Hadwiger transversal theorem in ℝd

Abstract

Hadwiger’s transversal theorem gives necessary and suffcient 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 ℝd in bijection with a set P of points in ℝd−1. Assume that there is a coloring of F with suffciently 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.

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Correspondence to Edgardo Roldán-Pensado.

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Supported by the Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (NRF-2010-0021048).

Research was conducted while visiting KAIST in Daejeon, South Korea.

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Holmsen, A.F., Roldán-Pensado, E. The colored Hadwiger transversal theorem in ℝd . Combinatorica 36, 417–429 (2016). https://doi.org/10.1007/s00493-014-3192-2

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Mathematics Subject Classification (2010)

  • 52A35
  • 52A20