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The following result was proved by Barany in 1982: For every d >= 1 there exists c_d > 0 such that for every n-point set S in R^d there is a point p in R^d contained in at least c_d n^{d+1} - O(n^d) of the simplices spanned by S.
A theorem on general measure
Richard Rado · 1947
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Equipartition of convex sets
Robert C. Buck and Ellen F. Buck · 1948
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Extremalaufgaben über endliche Punktsysteme
Franz Kárteszi · 1955
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A problem in geometric probability
James G. Wendel · 1962
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Jack G. Ceder · 1964
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Topics on Tournaments
John W. Moon · 1968
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Su un teorema di Kárteszi nella geometria combinatoria
Endre Boros and Zoltán Füredi · 1977
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A generalization of Carathéodory’s theorem
Imre Bárány · 1982
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The number of triangles covering the center of an n n -set
Endre Boros and Zoltán Füredi · 1984
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Edward R. Fadell and Sufian Husseini · 1988
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Noga Alon, Imre Bárány, Zoltán Füredi, and Daniel J. Kleitman · 1992
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Rade T. Živaljević and Siniša T. Vrećica · 1992
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David Eppstein · 1993
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Jiří Matoušek · 2002
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Shakhar Smorodinsky · 2003
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On k k -Sets and Applications
Ulrich Wagner · 2003
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Rade T. Živaljević · 2004
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Algorithms in real algebraic geometry
Saugata Basu, Richard Pollack, and Marie-Françoise Roy · 2006
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Boris Bukh · 2006
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Borsuk-Ulam type theorems on Stiefel manifolds
Akira Inoue · 2006
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Counting triangle crossings and halving planes
Tamal K. Dey and Herbert Edelsbrunner · 1994
Cited alongside, same era.
User’s guide to equivariant methods in combinatorics II
Rade T. Živaljević · 1998
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Entering and leaving j j -facets
Emo Welzl · 2001
Cited alongside, same era.
Eppstein’s bound on intersecting triangles revisited
Gabriel Nivasch and Micha Sharir
Cited in the paper.
Using the Borsuk–Ulam theorem
Jiří Matoušek · 2008
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