Generalized breadths, circular Cantor sets, and the least area UCC (Q1338462)

A universal convex cover (UCC for short) is a convex set in \(\mathbb{R}^ 2\) which, after shifting and rotating, can cover any point set of unit diameter. In this short and very readable paper, the author takes on an old question by Lebesgue, of determining the least area UCC. The smallest known UCC, of area slightly less than 0.8441 is provided by \textit{H. C. Hansen} [Geom. Dedicata 42, No. 2, 205-213 (1992; Zbl 0752.52011)]. This paper provides a simple construction (circular Cantor sets) improving the best known lower bound on the area of a UCC (with an area of slightly more than 0.8257). The paper's best construction gives a lower bound of \({3 \over 4} [\tan {\pi \over 6} + \sum_{i \geq 0} 2^ i \tan {\pi \over 18.3^ i}]\), which is slightly more than 0.8271.