Back in 2015, I reported some progress on this difficult problem in plane geometry. I’m happy to report some more.
First, remember the story. A subset of the plane has diameter 1 if the distance between any two points in this set is ≤ 1. A universal covering is a convex subset of the plane that can cover a translated, reflected and/or rotated version of every subset of the plane with diameter 1. In 1914, the famous mathematician Henri Lebesgue sent a letter to a fellow named Pál, challenging him to find the universal covering with the least area.
Pál worked on this problem, and 6 years later he published a paper on it. He found a very nice universal covering: a regular hexagon in which one can inscribe a circle of diameter 1. This has area
But he also found a universal covering with less area, by removing two triangles from this hexagon—for example, the triangles C1C2C3 and E1E2E3 here:
The resulting universal covering has area
In 1936, Sprague went on to prove that more area could be removed from another corner of Pál’s original hexagon, giving a universal covering of area
In 1992, Hansen took these reductions even further by removing two more pieces from Pál’s hexagon. Each piece is a thin sliver bounded by two straight lines and an arc. The first piece is tiny. The second is downright microscopic!
Hansen claimed the areas of these regions were 4 · 10-11 and 6 · 10-18. This turned out to be wrong. The actual areas are 3.7507 · 10-11 and 8.4460 · 10-21. The resulting universal covering had an area of
This tiny improvement over Sprague’s work led Klee and Wagon to write:
it does seem safe to guess that progress on [this problem], which has been painfully slow in the past, may be even more painfully slow in the future.
However, in 2015 Philip Gibbs found a way to remove about a million times more area than Hansen’s larger region: a whopping 2.233 · 10-5. This gave a universal covering with area
Karine Bagdasaryan and I helped Gibbs write up a rigorous proof of this result, and we published it here:
Greg Egan played an instrumental role as well, catching various computational errors.
At the time Philip was sure he could remove even more area, at the expense of a more complicated proof. Since the proof was already quite complicated, we decided to stick with what we had.
But this week I met Philip at The philosophy and physics of Noether’s theorems, a wonderful workshop in London which deserves a full blog article of its own. It turns out that he has gone further: he claims to have found a vastly better universal covering, with area
This is an improvement of 2.178245 × 10-5 over our earlier work—roughly equal to our improvement over Hansen.
You can read his argument here:
• Philip Gibbs, An upper bound for Lebesgue’s universal covering problem, 22 January 2018.
I say ‘claims’ not because I doubt his result—he’s clearly a master at this kind of mathematics!—but because I haven’t checked it and it’s easy to make mistakes, for example mistakes in computing the areas of the shapes removed.
It seems we are closing in on the final result; however, Philip Gibbs believes there is still room for improvement, so I expect it will take at least a decade or two to solve this problem… unless, of course, some mathematicians start working on it full-time, which could speed things up considerably.