A minimal Lebesgue covering exists (Q1099413)

A famous problem, due to Lebesgue, consists in determining the minimal area of covering figure \(\Phi\) for the family of all sets whose diameter is not greater than one. It is known that \(\Phi\) is not convex. A covering figure will be called irreducible, if no of its proper subsets is also a covering. The main result of the paper is the following Theorem. Any irreducible covering figure \(\Phi\) of minimal area is star-shaped; it is possible to choose the center of a polar system of coordiates inside \(\Phi\) so that its boundary \(\partial \Phi\) is given by the equation \(\tau =\rho(\phi)\), \(0\leq \phi \leq 2\pi\), where the function \(\rho(\phi)\) satisfies the inequalities: 1) \(c^ 2\leq \rho (\phi)\leq\leq 1-c^ 2;\) here \(c=1- 1/\sqrt{3},\) and 2) \(| \rho (\phi_ 1)-\rho (\phi_ 2)| \leq 5\cdot | \phi_ 1-\phi_ 2|\) for any \(\phi_ 1,\phi_ 2\in [0,2\pi]\) independent of the choice of the polar axis.