International Journal of Mathematics and Mathematical Sciences
Volume 22 (1999), Issue 2, Pages 401-410

Sections of simplices

Nagabhushana Prabhu

Purdue University, MGL 1303, West Lafayette 47907, IN, USA

Received 28 August 1995; Revised 10 October 1995

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 d/2kd, the relative interior of every k-face of a d-simplex Δd can be intersected by a 2(dk)-dimensional affine flat. Bezdek, Bisztriczky, and Connelly's results [2] show that the condition kd/2 above cannot be dropped and hence raise the question of determining, for all 0k,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),wherew=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+jd,M(j,k;d)f3j/41(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.