Improving the first selection lemma in R3

We present new bounds on the first selection lemma in R3 . This makes progress on the open problems of Bukh, Matoušek 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. Categories and Subject Descriptors G.2 [Combinatorics]: Counting Problems General Terms Theory Keywords Centerpoints, First Selection Lemma, Hitting Simplices, Location Depth

• Type: Conference
• Year: 2010
• Where: COMPGEOM
• Authors: Abdul Basit, Nabil H. Mustafa, Saurabh Ray, Sarfraz Raza
• Keywords: COMPGEOM 2010, Selection Lemma, Upper Bound, first Selection Lemma