An L^1 estimate for half-space discrepancy

For every unit vector $\sigma \in {\mathrm{\Sigma }}_{d-1}$ and every $r\ge 0$, let % % \begin{displaymath} P_{\sigma,r}=[-1,1]^d\cap{t\in\Rr^d:t\cdot\sigma\le r} \end{displaymath} % % denote the intersection of the cube $[-1,1{]}^d$ with a half-space containing the origin $0\in {\text{Rr}}^d$. We prove that if $N$ is the $d$-th power of an odd integer, then there exists a distribution $\text{PPP}$ of $N$ points in $[-1,1{]}^d$ such that % % \begin{displaymath} \sup_{r\ge0} \int_{\Sigma_{d-1}}\vert\card(\PPP\cap P_{\sigma,r})-N2^{-d} \vert P_{\sigma,r}\vert\vert,\dd\sigma \le c_d(\log N)^d, \end{displaymath} % % generalizing an earlier result of Beck and the first author.