Tetrahedral and pentahedral cages for discs

Abstract

This paper is about cages for compact convex sets. A cage is the 1-skeleton of a convex polytope in ℝ³. A cage is said to hold a set if the set cannot be continuously moved to a distant location, remaining congruent to itself and disjoint from the cage.

In how many “truly different” positions can (compact 2-dimensional) discs be held by a cage? We completely answer this question for all tetrahedra. Moreover, we present pentahedral cages holding discs in a large number (57) of positions.