The Holt-Klee condition for oriented matroids (Q1041605)

It has recently been shown by \textit{F. Holt} and \textit{V. Klee} [Contemp. Math. 223, 201--216 (1999; Zbl 0916.90206)] that every (generic) LP orientation of the graph of a \(d\)-polytope satisfies a directed version of the \(d\)-connectivity property, i.e., there are \(d\) internally disjoint directed paths from a unique source to a unique sink. By enforcing this property and its dual interpretation in terms of line shellings, two new classes of orientated matroids \(HK\) and \(HK^*\) are introduced in the paper under review. Both classes contain all representable oriented matroids by the Holt-Klee theorem. The authors provide a construction of an infinite family of non-\(HK^*\) oriented matroids, but it is not clear whether there exists any non-\(HK\) oriented matroid. The authors give a complete classification of the oriented matroids of rank 4 on 8-element ground set with respect to the \(HK\), \(HK^*\), Euclidean and Shannon properties. The classification shows that there exists no non-\(HK\) oriented matroid in this class.