Note on Lebesgue's universal cover problem

Abstract

A universal cover is a convex set that covers all sets of diameter 1 after some rotation, reflection, and translation. First, we show that a regular dodecahedron that circumscribes a ball of diameter 1 is not a universal cover in $\mathbb R^3$, answering a question of Chakerian [K]. Second, we improve the bounds on the minimum volume of a universal cover in $\mathbb R^3$ to $(0.545193,0.655984)$. Third, we give an infinite family of centrally symmetric polytopes in $\mathbb R^d$ with $d(d+1)$ facets that circumscribe the unit ball and are universal covers. This is the maximum possible number and proves a conjecture of Makeev [M2]. Finally, on the least number of facets of a centrally symmetric polytope in $\mathbb R^d$ that circumscribe the unit ball and is not a universal cover, we generalize Makeev's lower bound [M1] and improve his upper bound [M3].