Scribability problems for polytopes

TL;DR

This paper investigates classical and new scribability problems for polytopes, providing existence results for polytopes with faces tangent to or avoiding a sphere, advancing understanding of geometric realizability constraints.

Contribution

It solves classical scribability problems for stacked and cyclic polytopes, completes the classification of non-weakly circumscribable 3-polytopes, and introduces generalized $(i,j)$-scribability problems with new existence results.

Findings

Existence of non-scribable stacked and cyclic polytopes for all dimensions and face levels.

Construction of non-weakly circumscribable 3-polytopes.

Examples of polytopes with faces avoiding or cutting the sphere for cases where j-i ≤ d-3.

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 Gr\"unbaum 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 $j-i \le d-3$.