A short proof of the \(d\)-connectedness of \(d\)-polytopes (Q1343799)

A \(d\)-polytope is a \(d\)-dimensional set that is the convex hull of a finite set of points in \(E^ n\). The graph of a \(d\)-polytope is the graph formed by its vertices and edges. A graph is said to be \(d\)- connected provided it has at least \(d+ 1\) vertices and no two vertices can be separated by removing fewer than \(d\) vertices. One of the most fundamental theorems on convex polytopes is that the graph of a \(d\)- polytope is \(d\)-connected. This was originally proved by \textit{M. L. Balinski} [Pac. J. Math. 11, 431-434 (1961; Zbl 0103.396)]. Later another proof was found by the author [Isr. J. Math. 16, 62-72 (1973; Zbl 0273.52004)] and recently a new proof was given by \textit{A. Brøndsted} and \textit{G. Maxwell} [Can. Math. Bull. 32, No. 2, 252-254 (1989; Zbl 0751.52005)]. In this note we give a new proof that is very short and conceptually much simpler than the other known proofs.