Ray-shooting depth: computing statistical data depth of point sets in the plane
ESA'11 Proceedings of the 19th European conference on Algorithms
A proof of the Oja depth conjecture in the plane
Computational Geometry: Theory and Applications
| Hi-index | 0.00 |
The so-called first selection lemma states the following: given any set P of n points in ℝd , there exists a point in ℝd 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 ℝ3. In particular, we prove that c 3≥0.00227, improving the previous best result of c 3≥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 c 3≤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).