Hitting Simplices with Points in R
نویسندگان
چکیده
The so-called first selection lemma states the following: given any set P of n points in R, there exists a point in R contained in at least cdn − O(n) simplices spanned by P , where the constant cd depends on d. We present improved bounds on the first selection lemma in R. In particular, we prove that c3 ≥ 0.00227, improving the previous best result of c3 ≥ 0.00162 by Wagner [Wag03]. This makes progress, for the three dimensional case, on the open problems of Bukh, Matoušek and Nivasch [BMN10] (where it is proven that c3 ≤ 1/4 ≈ 0.00390) and Boros-Füredi [BF84] (where the two-dimensional case was settled).
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