Peeling the Grid

TL;DR

This paper proves that the number of convex layers in an n×n grid, obtained by iteratively removing convex hull vertices, is on the order of n^{4/3}, providing a tight bound on this geometric peeling process.

Contribution

It establishes a tight asymptotic bound of Θ(n^{4/3}) on the number of convex layers in an integer grid, advancing understanding of geometric peeling processes.

Findings

Number of convex layers is Θ(n^{4/3})

The process terminates after O(n^{4/3}) iterations

Provides a tight bound for convex hull peeling in grids

Abstract

Consider the set of points formed by the integer $n \times n$ grid, and the process that in each iteration removes from the point set the vertices of its convex-hull. Here, we prove that the number of iterations of this process is O(n^{4/3}); that is, the number of convex layers of the $n\times n$ grid is \Theta(n^{4/3}).