Scribability problems for polytopes

In this paper we study various scribability problems for polytopes. We begin with the classical k-scribability problem proposed by Steiner and generalized by Schulte, which asks about the existence of d-polytopes that cannot be realized with all k-faces tangent to a sphere. We answer this problem for stacked and cyclic polytopes for all values of d and k. We then continue with the weak scribability problem proposed by Grunbaum and Shephard, for which we complete the work of Schulte by presenting non weakly circumscribable 3-polytopes. Finally, we propose new (i, j)-scribability problems, in a strong and a weak version, which generalize the classical ones. They ask about the existence of d-polytopes that cannot be realized with all their i-faces "avoiding" the sphere and all their j-faces "cutting" the sphere. We provide such examples for all the cases where j - i <= d - 3. (c) 2017 Published by Elsevier Ltd.