Conway's fried potato problem revisited (Q1924940)

Assume we have two convex bodies \(K\) and \(B\) in the \(d\)-dimensional Euclidean space \(E^d\), and one of them, say \(K\), is sliced into \(n \geq 1\) pieces by \(n - 1\) successive hyperplane cuts (just one piece being divided by each cut). We determine a sharp lower bound for the size of the largest homothetic copy of the other convex body \(B\) (in terms of some parameters of \(K\) and \(B)\), which has a translate contained in one of the pieces of \(K\).