  The $L_p$-discrepancy is a classical quantitative measure for the
irregularity of distribution of an $N$-element point set in the $d$-dimensional
unit cube. Its inverse for dimension $d$ and error threshold $\varepsilon \in
(0,1)$ is the number of points in $[0,1)^d$ that is required such that the
minimal normalized $L_p$-discrepancy is less or equal $\varepsilon$. It is well
known, that the inverse of $L_2$-discrepancy grows exponentially fast with the
dimension $d$, i.e., we have the curse of dimensionality, whereas the inverse
of $L_{\infty}$-discrepancy depends exactly linearly on $d$. The behavior of
inverse of $L_p$-discrepancy for general $p \not\in \{2,\infty\}$ was an open
problem since many years. Recently, the curse of dimensionality for the
$L_p$-discrepancy was shown for an infinite sequence of values $p$ in $(1,2]$,
but the general result seemed to be out of reach. In the present paper we show
that the $L_p$-discrepancy suffers from the curse of dimensionality for all $p$
in $(1,\infty)$ and only the case $p=1$ is still open. This result follows from
a more general result that we show for the worst-case error of positive
quadrature formulas for an anchored Sobolev space of once differentiable
functions in each variable whose first mixed derivative has finite $L_q$-norm,
where $q$ is the H\"older conjugate of $p$.
