Long lines in subsets of large measure in high dimension

We show that for any set A⊆ [0 , 1] ⁿ with Vol (A) ≥ 1 / 2 there exists a line ℓ such that the one-dimensional Lebesgue measure of ℓ∩ A is at least Ω (n^{1 / 4}) . The exponent 1/4 is tight. More generally, for a probability measure μ on Rⁿ and 0 < a< 1 define L(μ,a):=infA;μ(A)=asupℓline|ℓ∩A| where | · | stands for the one-dimensional Lebesgue measure. We study the asymptotic behavior of L(μ, a) when μ is a product measure and when μ is the uniform measure on the ℓ_p ball. We observe a rather unified behavior in a large class of product measures. On the other hand, for ℓ_p balls with 1 ≤ p≤ ∞ we find that there are phase transitions of different types.