Scribability problems for polytopes

Abstract 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 .

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