Holding convex polyhedra by circular rings

Abstract

In 1995, T. Zamrescu proved that most convex bodies can be held by circles, that is, for most convex bodies $\mathcal{B}$ it is possible to attach a hinged circular ring of appropriate size to $\mathcal{B}$ so that it cannot slip out of $\mathcal{B}$. Since then, many results have been obtained concerning the existence of such circles for various convex polyhedra, and the sizes of such circles when they exist. It seems, however, that these results were obtained individually by ad hoc methods. In this paper we develop a unified concept and methods enabling a systematic presentation of these results, and we also obtain a few new results. A complete survey on the topic is also presented.