Upper bounds for stabbing simplices by a line

It is known that for every dimension d≥2 and every k<d there exists a constant c_d,k>0 such that for every n-point set X⊂R^d there exists a k-flat that intersects at least c_d,kn^d+1−k−o(n^d+1−k) of the (d−k)-dimensional simplices spanned by X. However, the optimal values of the constants c_d,k are mostly unknown. The case k=0 (stabbing by a point) has received a great deal of attention. In this paper we focus on the case k=1 (stabbing by a line). Specifically, we try to determine the upper bounds yielded by two point sets, known as the stretched grid and the stretched diagonal. Even though the calculations are independent of n, they are still very complicated, so we resort to analytical and numerical software methods. We provide strong evidence that, surprisingly, for d=4,5,6 the stretched grid yields better bounds than the stretched diagonal (unlike for all cases k=0 and for the case (d,k)=(3,1), in which both point sets yield the same bound). Our experiments indicate that the stretched grid yields c_4,1≤0.00457936, c_5,1≤0.000405335, and c_6,1≤0.0000291323.