Sections of simplices (Q1301933)

The author shows that the relative interiors of all the \(k\)-dimensional faces of a \(d\)-dimensional simplex, \( k\geq \lfloor d/2\rfloor,\) can be intersected by an affine flat of dimension \(2 (d - k).\) The result yields a counter-intuitive fact that the relative interiors of all the facets of any \(d\)-simplex can be intersected by a 2-dimensional plane. Bezdek, Bisztriczky, and Connelly's results [\textit{K. Bezdek, T. Bisztriczky}, and \textit{R. Connelly} [Monatsh. Math. 109, No. 1, 39-48 (1990; Zbl 0712.52012)], show that the restriction \( k\geq \lfloor d/2\rfloor\) above cannot be dropped and hence raise the question of determining, for all \(1\leq j, k < d,\) the function \(C(j,k;d),\) defined as the smallest number of \(j\)-flats needed to intersect the relative interiors of all the \(k\)-faces of a \(d\)-simplex. Using probabilistic arguments, the following fairly nontrivial upper bound on \(C(j,k;d)\) is proved. Let \(w =\min(\max(\lfloor {j}/{2}\rfloor+k,j),d).\) Then for \(1\leq k,j < d,\) \[ C(j,k;d)\leq{\binom{d+1}{k+1}}{\binom{w+1}{k+1}}^{-1} \log\binom{d+1}{k+1}. \] Another important problem is to determine, for all \(1\leq k,j < d,\) the function \(M (j, k; d),\) defined as the largest number of \(k\)-faces of a \(d\)-simplex whose relative interiors can be intersected by a \(j\)-flat. The author obtains the lower bound \(\binom{w+1}{k+1}\leq M (j, k; d)\) on \(M(j,k;d).\) For large \(d,\) \(d > d_0,\) and \(k+j\geq d,\) it is shown that the following is a tight upper bound on \(M(j,k;d):\) \(M(j,k;d)\leq f_{\lceil 3/4\rceil-1}(d+1,j),\) where \(f_m(n,q)\) is the number of \(m\)-faces in a cyclic \(q\)-polytope with \(n\)-vertices.