Let $F$ and $F_1,\ldots,F_n$ be bodies of constant width 1 in $\mathbb{R}^d$ such that $F_1,\ldots,F_n$ are pairwise disjoint and all intersect non-trivially (i.e. in at least one point) with $F$. What is the maximum cardinality of $n$? (of course, this will depend on $d$).
