Hitting Simplices with Points in $R^3$
The so-called first selection lemma states the following: given any set P of n points in a"e (d) , there exists a point in a"e (d) contained in at least c (d) n (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 a"e(3). In particular, we prove that c (3)a parts per thousand yen0.00227, improving the previous best result of c (3)a parts per thousand yen0.00162 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 c (3)a parts per thousand currency sign1/4(4)a parts per thousand 0.00390) and Boros and Furedi (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).