Long lines in subsets of large measure in high dimension

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.