Interior points of the convex hull of few points in \(\mathbb{E}^ d\) (Q1181780)

We show that if \(P\subset\mathbb{E}^ d\), \(| P|=d+k\), \(d\geq k\geq 1\) and \(O\in\hbox{int }\hbox{conv }P\), then there exists a simplex \(S\) of dimension \(\geq[d/k]\) with vertices in \(P\), satisfying \(O\in\hbox{rel int }S\), the bound being sharp. We give an upper bound for the minimal number of vertices of facets of a \((j-1)\)-neighbourly convex polytope in \(\mathbb{E}^ d\) with \(v\) vertices.