Sections of simplices

We show that for ⌊ d / 2 ⌋ ≤ k ≤ d , the relative interior of every k -face of a d -simplex Δ d 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 Δ d . Using probabilistic arguments, we show that C ( j , k ; d ) ≤ ( d + 1 k + 1 ) ( w + 1 k + 1 ) log ( d + 1 k + 1 ) , where w = min ( max ( ⌊ j 2 ⌋ + k , j ) , d ) . ( * ) Finally, we consider the function M ( j , k ; d ) , defined as the largest number of k -faces of Δ d 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 ⌈ 3 j / 4 ⌉ − 1 ( d + 1 , j ) , where f m ( 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.