The Small Ball Inequality is a conjectural lower bound on sums the L-infinity
norm of sums of Haar functions supported on dyadic rectangles of a fixed volume
in the unit cube. The conjecture is fundamental to questions in discrepancy
theory, approximation theory and probability theory. In this article, we
concentrate on a special case of the conjecture, and give the best known lower
bound in dimension 3, using a conditional expectation argument.