Lebesgue's universal covering problem on the minimum-area convex shape in the plane that can cover any shape of diameter one

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

For $A\subset\mathbb R^2$, its diameter is $\sup\{|x-y|:x,y\in A\}$. A convex universal cover is a convex planar set $K$ containing an isometric copy of every set $A$ of diameter one. An isometry may translate, rotate or reflect $A$. Area means two-dimensional Lebesgue measure.

Statement recorded in the supplied source.

Find a convex universal cover of minimum area and determine that minimum. [S2]

Short English statement

What is the smallest area of a convex planar set that can contain, after a rigid motion, every planar set of diameter one?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 7200072 (AMR-071-0072). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Philip Gibbs, A New Slant on Lebesgue’s Universal Covering Problem (2014); arXiv:1401.8217. https://arxiv.org/pdf/1401.8217.