Abdul Basit
ABSTRACT
We present new bounds on the first selection lemma in ℜ3. This makes progress on the open problems of Bukh, Matouaek and Nivash [6] and Boros-Füredi [4] for the three-dimensional case, improving the previously best result of Wagner [8]. While our results narrow the gap between the current best lower and upper bounds, they do not settle this question. However, they indicate that it is the current lower-bounds that are not tight, and we conjecture that the lower-bounds can be further improved to match the current upper bound.
- N. Alon, I. Bárány, Z. Füredi, and D. J. Kleitman. Point selections and weak e-nets for convex hulls. Combinatorics, Probability & Computing, 1:189--200, 1992.Google Scholar
Cross Ref
- N. Alon and D. Kleitman. Piercing convex sets and the hadwiger debrunner (p, q)-problem. Adv. Math., 96(1):103--112, 1992.Google ScholarCross Ref
- I. Barany. A generalization of caratheodory's theorem. Discrete Math., 40:189--200, 1982.Google ScholarDigital Library
- E. Boros and Z. Furedi. The number of triangles covering the center of an n-set. Geometriae Dedicata, 17(1):69--77, 1984.Google ScholarCross Ref
- B. Bukh. A point in many triangles. Electronic Journal of Combinatorics, 13(10), 2006.Google Scholar
- B. Bukh, M. Matousek, and G. Nivasch. Stabbing simplices by points and flats. http://arxiv.org/abs/0804.4464, 2008. Google ScholarDigital Library
- J. Matousek. Lectures in Discrete Geometry. Springer-Verlag, New York, NY, 2002. Google ScholarDigital Library
- U. Wagner. On k-Sets and Applications. PhD thesis, ETH Zurich, 2003.Google Scholar
Index Terms
- Improving the first selection lemma in R3
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