Long lines in subsets of large measure in high dimension (Q6070364)

A very interesting finding of the probability theory is established: for any set \(A\subseteq [0, 1]^n\) with \(\mathrm{Vol}(A)\geq 1/2\) there exists a line \(l\) such that the one-dimensional Lebesgue measure of \(l\cap A\) is at least \(\Omega(n^{1/4})\). The exponent 1/4 is tight. For a probability measure \(\mu\) on \(\mathbb R^n\) and \(0<a<1\) define \[ L(\mu, a):=\inf_{A;\mu(A)=a}\sup_{l \text{ line}} l\cap A. \] where \(|\cdot|\) stands for the one-dimensional Lebesgue measure. The asymptotic behavior of \(L(\mu, a)\) is studied. Very nice figures, well-developed theorems: this paper is an excellent application of the theory of probability.