Extremal Distributions of Discrepancy functions

The irregularities of a distribution of $N$ points in the unit interval are often measured with various notions of discrepancy. The discrepancy function can be defined with respect to intervals of the form $[0,t)\subset [0,1)$ or arbitrary subintervals of the unit interval. In the former case, it is a well known fact in discrepancy theory that the $N$-element point set in the with the lowest $L_2$ or $L_{\infty}$ norm of the discrepancy function is the centered regular grid $$\Gamma_N:=\left\{\frac{2n+1}{2N}: n=0,1,\dots,N-1\right\}.$$ We show a stronger result on the distribution of discrepancy functions of point sets in $[0,1]$, which basically says that the distribution of the discrepancy function of $\Gamma_N$ is in some sense minimal among all $N$-element point sets. As a consequence, we can extend the above result to rearrangement-invariant norms, including $L_p$, Orlicz and Lorentz norms. We study the same problem for the discrepancy notions with respect to arbitrary subintervals. In this case, we will observe that we have to deal with integrals of convolutions of functions. To this end, we prove a general upper bound on such expressions, which might be of independent interest as well.