Long lines in subsets of large measure in high dimension

We show that for any set $A\subseteq [0,1{]}^n$ with $\text{Vol}(A)\ge 1\mathrm{/}2$ there exists a line $\mathrm{\ell }$ such that the one-dimensional Lebesgue measure of $\mathrm{\ell }\cap A$ is at least $\mathrm{\Omega }(n^{1\mathrm{/}4})$. The exponent $1\mathrm{/}4$ 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 $\mathrm{\mid }\cdot \mathrm{\mid }$ stands for the one-dimensional Lebesgue measure. We study the asymptotic behavior of $L(\mu ,a)$when$\mu $isaproductmeasureandwhen$\mu $ is the uniform measure on the ${\mathrm{\ell }}_p$ ball. We observe a rather unified behavior in a large class of product measures. On the other hand, for ${\mathrm{\ell }}_p$ balls with $1 \leq p \leq \infty$ we find that there are phase transitions of different types.