Math Conjectures

Lebesgue's Universal Covering Problem

OPENMajorExact constant problemProposed 1914 · Standard version

Canonical statement

Let
U=inf⁡{area⁡(K):K⊂R2 is convex and every planar set S with diam⁡(S)≤1 has an isometric copy contained in K}. U=\inf\bigl\{\operatorname{area}(K): K\subset\mathbb R^2\text{ is convex and every planar set }S \text{ with }\operatorname{diam}(S)\leq1 \text{ has an isometric copy contained in }K\bigr\}.
Determine the exact value of UU, and whether the infimum is attained.
View source LaTeX
Let
\[
  U=\inf\bigl\{\operatorname{area}(K):
    K\subset\mathbb R^2\text{ is convex and every planar set }S
    \text{ with }\operatorname{diam}(S)\leq1
    \text{ has an isometric copy contained in }K\bigr\}.
\]
Determine the exact value of \(U\), and whether the infimum is attained.
DG-015:[Lebesgue1914Universal][BrassMoserPach2005][Gibbs2018Universal]
Discrete geometryGeometry and topology

Notes

Lebesgue asked in 1914 for the convex set of least area containing an isometric copy of every planar set of diameter at most one — a universal cover — together with the question of whether a least-area cover actually exists [Lebesgue1914Universal]. Formally, one seeks the infimum UU of the areas of convex universal covers, and whether it is attained.

A regular hexagon of width one is a classical universal cover, of area 3/2≈0.866\sqrt3/2\approx0.866, and a century of constructions has proceeded by carefully trimming material from it. The best known upper bound now lies below 0.8450.845, with the latest improvement due to Gibbs [Gibbs2018Universal]. Lower bounds for the area of any universal cover have also improved over time, but they remain clearly separated from the best constructions; the history and state of the problem are surveyed by Brass, Moser and Pach [BrassMoserPach2005].

No extremal universal cover has been characterized, and even whether the infimum is attained is unsettled. The problem is open, and a resolution would require constructions and lower bounds meeting at a common value UU.

References (3)

The boxed statement is the canonical open formulation — not a stronger variant or a related research program. The status reflects the catalog's last review; do your own literature search before investing serious effort.