How to hold a convex body? (Q1900021)

Let \({\mathcal B}\) be the class of all convex bodies in \(\mathbb{R}^3\) and \({\mathcal C}\) the class of all circles in \(\mathbb{R}^3\), both endowed with the Hausdorff metric \(\delta\). A convex body \(B\) is held by a circle \(C\) provided that \(C \cap \text{int} B = \emptyset\) and for some \(m \in N\) there is no continuous mapping \(f : [0,1] \to {\mathcal C}\) such that \(f(0) = C\), \(\delta (f(0), f(1)) > m\), and, for every \(t \in [0,1]\), \(f(t)\) is congruent with \(C\) and \(f(t) \cap \text{int} B = \emptyset\). The author proves that some convex bodies are held by circles; moreover, the class of convex bodies in \(\mathbb{R}^3\) which cannot be held by a circle is nowhere dense in \({\mathcal B}\).