Copyright © 1999 Nagabhushana Prabhu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We show that for , the relative interior of every -face of a -simplex can be intersected by a -dimensional affine flat. Bezdek, Bisztriczky, and Connelly's results  show that the condition above cannot be dropped and hence raise the question of determining, for all , an upper bound on the function , defined as the smallest number of -flats, , needed to intersect the relative interiors of all the -faces of . Using probabilistic arguments, we show that
Finally, we consider the function , defined as the largest number of -faces of whose relative interiors can be intersected by a -flat. We show that, for large and for all such that , where is the number of -faces in a cyclic -polytope with -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.