The extremal spheres theorem

Consider a polygon PP and all neighboring circles (circles going through three consecutive vertices of PP). We say that a neighboring circle is extremal if it is empty (no vertices of PP inside) or full (no vertices of PP outside). It is well known that for any convex polygon there exist at least two empty and at least two full circles, i.e. at least four extremal circles. In 1990 Schatteman considered a generalization of this theorem for convex polytopes in dd-dimensional Euclidean space. Namely, he claimed that there exist at least 2d2d extremal neighboring spheres for generic polytopes. His proof is based on the Bruggesser–Mani shelling method.In this paper, we show that there are certain gaps in Schatteman’s proof. We also show that using the Bruggesser–Mani–Schatteman method it is possible to prove that there are at least d+1d+1 extremal neighboring spheres. However, the existence problem of 2d2d extremal neighboring spheres is still open.