On asymptotic Lebesgue's universal covering problem

Abstract

Universal cover in ${\mathbb{E}}^n$ is a measurable set that contains a congruent copy of any set of diameter 1. Lebesgue's universal covering problem, posed in 1914, asks for the convex set of smallest area that serves as a universal cover in the plane ($n=2$). A simple universal cover in ${\mathbb{E}}^n$ is provided by the classical theorem of Jung, which states that any set of diameter 1 in an $n$-dimensional Euclidean space is contained in a ball $J_n$ of radius $\sqrt{\frac{n}{2n+2}}$; in other words, $J_n$ is a universal cover in ${\mathbb{E}}^n$. We show that in high dimensions, Jung's ball $J_n$ is asymptotically optimal with respect to the volume, namely, for any universal cover $U\subset {\mathbb{E}}^n$, $$\mathrm{Vol}(U)\ge (1-o(1){)}^n\mathrm{Vol}(J_n).$$