Lower Bounds on Strip Discrepancy for Nonatomic Colorings

Abstract.

 A method of proof is given for obtaining lower bounds on strip discrepancy when the distributions do not have atoms. Partition the unit square into an chessboard of congruent square pixels, where n is even. Color of the pixels red, and the rest blue. For any convex set A, let be the difference between the amounts of red and blue areas in A. Under a technical local balance condition, we prove there must be a strip S, of width less than , for which , where c is a positive constant, independent of n and the coloring. The proof extends methods discovered by Alexander and further developed by Chazelle, Matoušek, and Sharir. Integral geometric notions figure prominently.