Disjoint faces of complementary dimension (Q1878951)

This is a short note that proves: If \(P\) is a \(d\)-polytope that is not a simplex, then for all \(0<k<d,P\) has a \(k\)-face and a \((d-k)\)-face that are disjoint. The \(k=1\) case has been shown in [\textit{E. Miller} and \textit{D. Helm}, Pac. J. Math. 209, No. 1, 41--66 (2003; Zbl 1078.13009)]. The proof is a straightforward induction, breaking into two cases, where \(P\) has a facet that is not a simplex, or where \(P\) has a vertex figure that is not a simplex. The author also asks what can be said when some \(k\)-face of a polytope intersects all \((d-k)\)-faces.