A tight bound on the number of geometric permutations of convex fat objects in {\huge $\mathbf{\reals^d}$}

Abstract

We show that the maximum number of geometric permutations of a set of $n$ pairwise-disjoint convex and fat objects in $\reals^d$ is $O(n^{d-1})$. This generalizes the bound of $\Theta (n^{d-1})$ obtained by Smorodinsky et al. \cite{ssm98} on the number of geometric permutations of $n$ pairwise-disjoint balls.