Sections of simplices - Nagabhushana Prabhu (International Journal of Mathematics and Mathematical Sciences, 1999)
We show that for ⌊d/2⌋≤k≤d, the relative interior of every k-face of a d-simplex Δd can be intersected by a 2(d−k)-dimensional affine flat. Bezdek, Bisztriczky, and Connelly's results [2] show that the condition k≥⌊d/2⌋ above cannot be dropped and hence raise the question of determining, for all 0≤k,j<d, an upper bound on the function c(j,k;d), defined as the smallest number of j-flats, j<d, needed to intersect the relative interiors of all the k-faces of Δd. Using probabilistic arguments, we show that C( j,k;d)≤(d+1k+1)(w+1k+1)log(d+1k+1),         where  w=min(max(⌊j2⌋+k,j),d).                            (*)