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 Grünbaum 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 can not 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 $j-i\le d-3$.

25 pages, 11 figures. v2: minor changes

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Cited by in corpus (2)

• Continuous Maps from Spheres Converging to Boundaries of Convex Hulls
• A polytopal generalization of Apollonian packings and Descartes' theorem