Hitting Simplices with Points in ℝ³

The so-called first selection lemma states the following: given any set P of n points in ℝ^d, there exists a point in ℝ^d contained in at least c_dn^d+1-O(n^d) simplices spanned by P, where the constant c_d depends on d. We present improved bounds on the first selection lemma in ℝ³. In particular, we prove that c₃≥0. 00227, improving the previous best result of c₃≥0. 00162 by Wagner (On k-sets and applications. Ph. D. thesis, ETH Zurich, 2003). This makes progress, for the three-dimensional case, on the open problems of Bukh et al. (Stabbing simplices by points and flats. Discrete Comput. Geom., 2010) (where it is proven that c₃≤1/4⁴≈0. 00390) and Boros and Füredi (The number of triangles covering the center of an n-set. Geom. Dedic. 17(1):69-77, 1984) (where the two-dimensional case was settled).