Barnabas Janzer

Rotations inside convex Kakeya sets

Let K be a convex body in R^d that contains a copy of another body S in every possible orientation. Is it always possible to continuously move any one copy of S into another, inside K? This question was asked by Croft. We prove that the answer is positive if S is a line segment, but, surprisingly, the answer is negative in dimensions at least four for general S.