  We show that for any set $A\subseteq [0,1]^n$ with $\text{Vol}(A)\ge 1/2$
there exists a line $\ell $ such that the one-dimensional Lebesgue measure of
$\ell \cap A$ is at least $\Omega ( n^{1/4} )$. The exponent $1/4$ is tight.
More generally, for a probability measure $\mu $ on $\mathbb R ^n$ and $0<a<1$
define \begin{equation*} L(\mu ,a):= \inf_{A ; \mu(A) = a} \sup _{\ell \text{
line}} \big| \ell \cap A\big| \end{equation*} where $|\cdot | $ stands for the
one-dimensional Lebesgue measure. We study the asymptotic behavior of $L(\mu
,a)$ when $\mu $ is a product measure and when $\mu $ is the uniform measure on
the $\ell _p$ ball. We observe a rather unified behavior in a large class of
product measures. On the other hand, for $\ell_p$ balls with $1 \leq p \leq
\infty$ we find that there are phase transitions of different types.
