Gale diagrams of convex polytopes and positive spanning sets of vectors (Q797150)

A positive k-spanning set (PkSS) in Euclidean m-space \({\mathbb{R}}^ m\) is a set of points such that each open half-space of \({\mathbb{R}}^ m\) bounded by a hyperplane through the origin contains at least k points of the set. In this paper, the author investigates the maximal cardinality of a minimal (with respect to inclusion) PkSS of \({\mathbb{R}}^ m\), initially conjecturing that the number is 4km. Through Gale diagrams, this is associated in case \(k=2\) with the problem of determining the minimum number of vertices of an unneighbourly d-polytope, which is such that no vertex is joined to every other one by an edge. In a note at the end, the author announces an unneighbourly 36-polytope with 49 vertices, which leads to a counterexample to the conjecture for \(k=2\) and \(m\geq 12\).