Sharp Bounds For The Layer Number of Integer Grids

Abstract

The layer number of a finite point set is the number of iterations needed to delete it by repeatedly removing the vertices of its convex hull. Ambrus, Hsu, Peng, and Yan conjectured that the layer number of the $d$-dimensional integer grid $\{1,\ldots,n\}^d$ is of order $n^{2d/(d+1)}$ for every fixed $d$. We prove this conjecture. Let $P_i$ be the convex hull of the point set remaining after $i$ steps, and let $Z_n$ be the convex hull of the lattice points in the Euclidean ball of radius $n$. For every step that leaves a nonempty point set, the Minkowski sum $P_{i+1}+Z_n$ contains no vertex of $P_i+Z_n$. Integrality of normalized lattice volume, together with the Bárány--Larman vertex estimate for $Z_n$, gives a lower bound, independent of $i$, on the resulting volume decrease. Summing over $i$ yields the matching upper bound, even when $P_i$ is lower-dimensional. For $d\ge2$, the same upper bound holds uniformly over all nonempty subsets of $\{1,\ldots,n\}^d$.