On asymptotic Lebesgue's universal covering problem
Andrii Arman, Andriy Bondarenko, Andriy Prymak, Danylo Radchenko
TL;DR
The article proves that Jung's ball J_n is asymptotically volume-optimal as a universal cover in high dimensions, establishing a near-tight lower bound on the volume of any universal cover in terms of Vol(J_n) with subexponential corrections. It achieves this via a probabilistic-discretization approach: thickening minimal covers, discretizing the space of congruent copies, and employing a measurable-graph framework to force a contradiction unless volumes match the bound. The work links to Borsuk's problem, showing limits on deriving partition-based improvements from universal covers, and also clarifies asymptotic optimality across other geometric measures such as mean width and quermassintegrals, with translative results aligning with Jung's ball in the large-dimension limit.
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 $\sqrt{\tfrac{n}{2n+2}}$; 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 $U \subset \mathbb{E}^n$, $$ {\rm Vol}(U) \ge (1-o(1))^n{\rm Vol}(J_n). $$