Upper bounds for stabbing simplices by a line

It is known that for every dimension d ≥ 2 and k < there exists a constant c , > 0 such n -point set X ⊂ R -flat intersects at least + 1 − o ( ) of the -dimensional simplices spanned by . However, optimal values constants are mostly unknown. The case = (stabbing point) has received great deal attention. In this paper we focus on line). Specifically, try to determine upper bounds yielded two point sets, as stretched grid diagonal Even though calculations independent they still very complicated, so resort analytical numerical software methods. We provide strong evidence that, surprisingly, 4 5 6 yields better than (unlike all cases 3 in which both sets yield same bound). Our experiments indicate ≤ 00457936 000405335 0000291323