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). (*)Finally, we consider the function M(j,k;d), defined as the largest number of k-faces of Δd whose relative interiors can be intersected by a j-flat. We show that, for large d and for all k such that k+j≥d,M(j,k;d)≤f⌈3j/4⌉−1(d+1,j), where fm(n,q) is the number of m-faces in a cyclic q-polytope with n-vertices. Our results suggest a conjecture about face-lattices of polytopes that if proved, would play a useful role in further studies on sections of polytopes.
