  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 2d
extremal neighboring spheres. In this paper, we show that there are certain
gaps in Schatteman's proof, which is based on the Bruggesser-Mani shelling
method. We show that using this method it is possible to prove that there are
at least d+1 extremal neighboring spheres. However, the existence problem of 2d
extremal neighboring spheres is still open.
