Holding Circles and Fixing Frames

A circle C holds a convex body K ⊂ R3 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 ⊂ R3 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.