Holding Circles and Fixing Frames

摘要

A circle (C) holds a convex body (K subset mathbb {R}^3) if (K) can’t be moved far away from its position without intersecting (C). One of our results says that there is a convex body (K subset mathbb {R}^3) such that the set of radii of all circles holding (K) has infinitely many components. Another result says that the circle is unique in the sense that every frame different from the circle holds a convex body (K) (actually a tetrahedron) so that every nontrivial rigid motion of (K) intersects the frame.