Let P be a set of n points in \({\mathbb {R}}^d\) , in general position. We remove all of them one by one, in each step erasing one vertex of the convex hull of the current remaining set. Let \(g_d(P)\) denote the number of different removal orders we can attain while erasing all points of P this way, and let \(g_d(n)\) be the minimum of \(g_d(P)\) over all n-element point sets \(P\subset {\mathbb {R}}^d\) . Dumitrescu and Tóth showed that \(g_d(n)\le (d+1)^{(d+1)^2n}\) . We substantially improve their bound, by proving that \(g_d(n)=O((d+d\ln {d})^{(2+\frac{(d-1)}{\lfloor d\ln {d}\rfloor })n})\) . It follows that, for any \(\epsilon >0\) , there exist sufficiently high dimensional point sets \(P\subset {\mathbb {R}}^d\) with \(g_d(P)\le O(d^{(2+\epsilon )n})\) . This almost closes the gap between the upper bound and the best-known lower bound \((d+1)^n\) for large values of d.
