Let \(K\) be a proper convex body in \(\mathbb R^{3}\). Consider the space \({\mathcal C}_K\) of all circles in \(\mathbb R^{3}\) disjoint from \(\text{int}~K\). A circle holds \(K\) if it belongs to a bounded component of \({\mathcal C}_K(r)\) for some \(r\), where \({\mathcal C}_K(r)\) is the set of all circles in \({\mathcal C}_K\) of radius \(r\). Let \({\mathcal H}_K\subseteq {\mathcal C}_K\) denote the space of all holding circles of \(K\).
Theorem. Let \(n\in\mathbb N\). For all convex bodies \(K\), except for a nowhere dense family, \({\mathcal H}_K\) has at least \(n\) components.
A nice consequence of this result is as follows:
Corollary. For most bodies \(K\), \({\mathcal H}_K\) has infinitely many components.