An L¹ estimate for half-space discrepancy

For every unit vector $\sigma\in\Sigma_{d-1}$ and every $r\ge0$, 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 $0\in\Rr^d$. We prove that if $N$ is the $d$-th power of an odd integer, then there exists a distribution $\PPP$ 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.