Peeling the Grid

Consider the set of points formed by the integer n × 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 ( n4/3 ) ; that is, the number of convex layers of the n× n grid is Θ ( n4/3 ) . ∗Department of Computer Science; University of Illinois; 201 N. Goodwin Avenue; Urbana, IL, 61801, USA; [email protected]; http://www.uiuc.edu/~sariel/. Work on this paper was partially supported by a NSF AF award CCF-0915984. †Department of Mathematical Sciences; University of Illinois; 1409 W. Green St.; Urbana, IL, 61801, USA; [email protected]. 1 ar X iv :1 30 2. 32 00 v1 [ cs .D M ] 1 3 Fe b 20 13 Figure 1: A point set and its decomposition into convex layers.