New estimates for convex layer numbers

Starting with a finite point set X⊂Rd, the peeling process repeatedly removes the set of the vertices of the convex hull of the current set. The number of peeling steps required to completely remove X is called the layer number of X, denoted by L(X). In the article, we study the layer number of evenly distributed families of point sets contained in Bd, the d-dimensional unit ball. These sets consist of points in Bd whose minimal distance is asymptotically as large as possible. We show that for a set X belonging to an evenly distributed family, L(X)≥Ω(|X|1/d) holds, with the bound being asymptotically sharp. On the other hand, building on earlier results, we prove that L(X)≤O(|X|2/d) holds for d≥2, which improves greatly on the current upper bound of O(|X|(d+1)/2d) for d≥3. Finally, we provide a recursive construction of evenly distributed families whose sets satisfy L(X)=Θ(|X|2/d−1/(d2d−1)), showing that our upper bound is nearly tight.