Barnabás Janzer: Rotation inside convex Kakeya sets

Let K be a convex body (a compact convex set) 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? As a stronger question, is it always possible to continuously select, for each orientation, one copy of S in that orientation? In this talk we answer these questions of Croft.