First, remember the story. A subset of the plane has

diameter 1if the distance between any two points in this set is ≤ 1. Auniversal coveringis 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.

But you raise a very important point! I believe the two problems are not the same: if you allow yourself to reflect your sets of diameter 1, you can fit them in a convex set with smaller area. In the paper Gibbs wrote with Bagdasaryan and me, we really needed to allow reflections to obtain the shape we did. I believe Gibbs’ new paper also uses reflections.

I have tried to edit all 3 articles so they only mention the version that allows reflections, since that’s what we really consider. But both problems are interesting.

It’s also interesting to consider the version that doesn’t demand convexity. I discussed this alternative in Part 1.

]]>When something shows up, I’ll let everyone know about it on this blog.

]]>