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We study the extreme and the periodic $L_p$ discrepancy of point sets in the$d$-dimensional unit cube. The extreme discrepancy uses arbitrary sub-intervalsof the unit cube as test sets, whereas the periodic discrepancy is based onperiodic intervals modulo one. This is in contrast to the classical stardiscrepancy, which uses exclusively intervals that are anchored in the originas test sets. In a recent paper the authors together with Aicke Hinrichsstudied relations between the $L_2$ versions of these notions of discrepancyand presented exact formulas for typical two-dimensional quasi-Monte Carlopoint sets. In this paper we study the general $L_p$ case and deduce the exactorder of magnitude of the respective minimal discrepancy in the number $N$ ofelements of the considered point sets, for arbitrary but fixed dimension $d$,which is $(\log N)^{(d-1)/2}$.

Point sets with optimal order of extreme and periodic discrepancy