On asymptotic Lebesgue's universal covering problem

TL;DR

This paper demonstrates that in high-dimensional Euclidean spaces, Jung's classical universal cover is asymptotically optimal in volume among all universal covers, confirming its near-minimality as the dimension grows.

Contribution

The paper proves that Jung's ball provides an asymptotically optimal universal cover in high dimensions with respect to volume.

Findings

Jung's ball is asymptotically volume-minimizing in high dimensions.

Universal covers in high dimensions cannot have significantly smaller volume than Jung's ball.

The result confirms the near-optimality of Jung's cover as dimension increases.

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 $U \subset \mathbb{E}^n$, $$ {\rm Vol}(U) \ge (1-o(1))^n{\rm Vol}(J_n). $$