The extremal spheres theorem

Consider a polygon P and all neighboring circles (circles going through three consecutive vertices of P ). We say that a neighboring circle is extremal if it is empty (no vertices of P inside) or full (no vertices of P 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 d -dimensional Euclidean space. Namely, he claimed that there exist at least 2 d extremal neighboring spheres for generic polytopes. His proof is based on the Bruggesser–Mani shelling method. s 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 + 1 extremal neighboring spheres. However, the existence problem of 2 d extremal neighboring spheres is still open.