I try to focus on serious problems in this blog, mostly environmental issues and the attempt to develop ‘green mathematics’. But I seem unable to resist talking about random fun stuff now and then. For example, the Lebesgue universal covering problem. It’s not important, it’s just strange… but for some reason I feel a desire to talk about it.
For starters, let’s say the diameter of a region in the plane is the maximum distance between two points in this region. You all know what a circle of diameter 1 looks like. But an equilateral triangle with edges of length 1 also has diameter 1:
After all, two points in this triangle are farthest apart when they’re at two corners. And you’ll notice, if you think about it, that this triangle doesn’t fit inside a circle of diameter 1:
In 1914, the famous mathematician Henri Lebesgue sent a letter to a pal named Pál. And in this letter he asked: what’s the smallest possible region that contains every set in the plane with diameter 1?
He was actually more precise. The phrase “smallest possible” is vague, but Lebesgue wanted the set with the least possible area. The phrase “contains” is also vague, at least as I used it. I didn’t mean that our region should literally contain every set with diameter 1. Only the whole plane does that! I meant that we can rotate or translate any set with diameter 1 until it fits in our region. So a more precise statement is:
What is the smallest possible area of a region in the plane such that every set of diameter 1 in the plane can be rotated and translated to fit inside ?
You see why math gets a reputation of being dry: sometimes when you make a simple question precise it gets longer.
Even this second version of the question is a bit wrong, since it’s presuming there exists a region with smallest possible area that does the job. Maybe you can do it with regions whose area gets smaller and smaller, closer and closer to some number, but you can’t do it with a region whose area equals that number! Also, the word ‘region’ is a bit vague. So if I were talking to a nitpicky mathematician, I might pose the puzzle this way:
What is the greatest lower bound of the measures of closed sets such that every set of diameter 1 can be rotated and translated to fit inside ?
Anyway, the reason this puzzle is famous is not that it gets dry when you state it precisely. It’s that it’s hard to solve!
It’s called Lebesgue’s universal covering problem. A region in the plane is called a universal covering if it does the job: any set in the plane of diameter 1 can fit inside it, in the sense I made precise.
Pál set out to find universal coverings, and in 1920 he wrote a paper on his results. He found a very nice one: a regular hexagon circumscribed around a circle of diameter 1. Do you see why you can fit the equilateral triangle with side length 1 inside this?
This hexagon has area
But in the same paper, Pál showed you could safely cut off two corners of this hexagon and get a smaller universal covering! This one has area
He guessed this solution was optimal.
However, in 1936, a guy named Sprague sliced two tiny pieces off Pál’s proposed solution and found a universal covering with area
Here’s the idea:
The big hexagon is Pál's original solution. He then inscribed a regular 12-sided polygon inside this, and showed that you can remove two of the corners, say and and get a smaller universal covering. But Sprague noticed that near you can also remove the part outside the circle with radius 1 centered at , as well as the part outside the circle with radius 1 centered at
Sprague conjectured that this is the best you can do.
But in 1975, Hansen showed you could slice off very tiny corners off Sprague’s solution, each of which reduces the area by just
I think this microscopic advance, more than anything else, convinced people that Lebesgue’s universal covering problem was fiendishly difficult. Viktor Klee, in a parody of the usual optimistic prophecies people like to make, wrote that:
… progress on this question, which has been painfully slow in the past, may be even more painfully slow in the future.
Indeed, my whole interest in this problem is rather morbid. I don’t know any reason that it’s important. I don’t see it as connected to lots of other beautiful math. It just seems astoundingly hard compared to what you might initially think. I admire people who work on it in the same way I admire people who decide to ski across the Antarctic. It’s fun to read about—from the comfort of my living room, sitting by the fire—but I can’t imagine actually doing it!
In 1980, a guy named Duff did a lot better. All the universal coverings so far were convex subsets of the plane. But he considered nonconvex universal coverings and found one with area:
This changed the game a lot. If we only consider convex universal coverings, it’s possible to prove there’s one with the least possible area—though we don’t know what it is. But if we allow nonconvex ones, we don’t even know this. There could be solutions whose area gets smaller and smaller, approaching some nonzero value, but never reaching it.
So at this point, Lebesgue’s puzzle split in two: one for universal coverings, and one for convex universal coverings. I’ll only talk about the second one, since I don’t know of further progress on the first.
Remember how Hansen improved Sprague’s universal covering by chopping off two tiny pieces? In 1992 he went further. He again sliced two corners off Sprague’s solution. One, the same shape as before, reduced the area by just But the other was vastly larger: it reduced the area by a whopping
I believe this is the smallest convex universal covering known so far.
But there’s more to say. In 2005, Peter Brass and Mehrbod Sharifi came up with a nice lower bound. They showed any convex universal covering must have an area of least
They got this result by a rigorous computer-aided search for the convex set with the smallest area that contains a circle, equilateral triangle and pentagon of diameter 1.
If you only want your convex set to contain a circle and equilateral triangle of diameter 1, you can get away with this:
This gives a lower bound of
But the pentagon doesn’t fit in this set! Here is the least-area convex shape that also contains a pentagon of diameter 1, as found by Brass and Sharifi:
You’ll notice the triangle no longer has the same center as the circle!
To find this result, it was enough to keep the circle fixed, translate the triangle, and translate and rotate the pentagon. So, they searched through a 5-dimensional space, repeatedly computing the area of the convex hull of these three shapes to see how small they could make it. They considered 53,118,162 cases. Of these, 655,602 required further care—roughly speaking, they had to move the triangle and pentagon around slightly to see if they could make the area even smaller.
They say they could have done better if they’d also included a fourth shape, but this was computationally infeasible, since it would take them from a 5-dimensional search space to an 8-dimensional one. It’s possible that one could tackle this using a distributed computing project where a lot of people contribute computer time, like they do in the search for huge prime numbers.
If you hear of more progress on this issue, please let me know! I hope that sometime—perhaps tomorrow, perhaps decades hence—someone will report good news.
Hansen’s record-holding convex universal cover is here:
• H. Hansen, Small universal covers for sets of unit diameter, Geometriae Dedicata 42 (1992), 205–213.
It’s quite readable. This is where I got the picture of Pál’s solution and Sprague’s improvement.
The paper on the current best lower bound for convex universal coverings is also quite nice:
• Peter Brass and Mehrbod Sharifi, A lower bound for Lebesgue’s universal cover problem, International Journal of Computational Geometry & Applications 15 (2005), 537–544.
The picture of the equilateral triangle in the circle comes from an earlier paper, which got a lower bound of
by considering the circle and regular -gons of diameter 1 for all :
• Gy. Elekes, Generalized breadths, circular Cantor sets, and the least area UCC, Discrete & Computational Geometry 12 (1994), 439–449.
I have not yet managed to get ahold of Duff’s paper on the record-holding nonconvex universal covering:
• G. F. D. Duff, A smaller universal cover for sets of unit diameter, C. R. Math. Acad. Sci. 2 (1980), 37–42.
One interesting thing is that in 1963, Eggleston found a universal covering that’s minimal in the following sense: no subset of it is a universal covering. However, it doesn’t have the least possible area!
• H. G. Eggleston, Minimal universal covers in Israel J. Math. 1 (1963), 149–155.
Later, Kovalev showed that any ‘minimal’ universal covering in this sense is a star domain:
• M. D. Kovalev, A minimal Lebesgue covering exists (Russian), Mat. Zametki 40 (1986), 401–406, 430.
This means there’s a point inside the set such that for any other point in the set, the line segment connecting and is in the set. Any convex set is a star domain, but not conversely: