On asymptotic Lebesgue's universal covering problem

2025/12/03 by Andrii Arman, Andriy Bondarenko, Arman, Andrii +5
Mathematics · #49Q20 #52A22 #52A38 #52A39 #52A40 #Analytic and geometric function theory #Combinatorics (math.CO) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Metric Geometry (math.MG) #Point processes and geometric inequalities #Primary 52C17 #Secondary 52A20

paper · pdf · doi:10.48550/arxiv.2512.04023

openalex publication_date 2025/12/03 · openalex created_date 2025/12/05 · openalex updated_date 2026/07/30

Abstract

Universal cover in 𝔼ⁿ 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 𝔼ⁿ 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 √\tfracn2n+2; in other words, J_n is a universal cover in 𝔼ⁿ. 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 ⊂ 𝔼ⁿ, \rm Vol(U) ≥ (1-o(1))ⁿ\rm Vol(J_n).

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