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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References
J. L. Arocha, J. Bracho and L. Montejano: A colorful theorem on transversal lines to plane convex sets, Combinatorica 28 (2008), 379–384.
I. Bárány: A generalization of Carathéodory’s theorem, Discrete Math. 40 (1982), 141–152.
I. Bárány, A. F. Holmsen and R. Karasev: Topology of geometric joins, Discrete Comput. Geom. 53 (2015), 402–413.
A. BjÖrner, B. Korte and L. Lovász: Homotopy properties of greedoids, Adv. in Appl. Math. 6 (1985), 447–494.
A. Björner, M. Las Vergnas, B. Sturmfels, N. White and G. M. Ziegler: Oriented matroids, Encyclopedia of Mathematics and its Applications, Cambridge University Press, second edition, 1999.
C. CarathÉodory: Über den Variabilitätsbereich der Koežienten von Potenzreihen, die gegebene Werte nicht annehmen, Math. Ann. 64 (1907), 95–115.
J. Eckhoff: Helly, Radon, and Carathéodory type theorems, Handbook of convex geometry, Vol. A, B, North-Holland, Amsterdam, 1993, 389–448.
J. E. Goodman and R. Pollack: Hadwiger's transversal theorem in higher dimensions, J. Amer. Math. Soc. 1 (1988), 301–309.
H. Hadwiger: Ueber Eibereiche mit gemeinsamer Treffgeraden, Portugal. Math. 16 (1957), 23–29.
E. Helly: Über Systeme linearer Gleichungen mit unendlich vielen Unbekannten, Monatsh. Math. Phys. 31 (1921), 60–91.
G. Kalai and R. Meshulam: A topological colorful Helly theorem, Adv. Math. 191 (2005), 305–311.
Katchalski and Meir: Thin sets and common transversals, J. Geom. 14 (1980) 103–107.
P. Kirchberger: Über Tchebychefsche Annäaherungsmethoden, Math. Ann. 57 (1903), 509–540.
J. Matoušek: Lectures on discrete geometry, Graduate Texts in Mathematics, vol. 212, Springer-Verlag, New York, 2002.
P. Kirchberger: Using the Borsuk-Ulam theorem, Universitext, Springer-Verlag, Berlin, 2003.
R. Pollack and R. Wenger: Necessary and suffcient conditions for hyperplane transversals, Combinatorica 10 (1990), 307–311.
H. Tverberg: A generalization of Radon's theorem, J. London Math. Soc. 41 (1966), 123–128.
R. Wenger: A generalization of Hadwiger's transversal theorem to intersecting sets, Discrete Comput. Geom. 5 (1990), 383–388.
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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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DOI: https://doi.org/10.1007/s00493-014-3192-2