I’m at the Banff International Research Station this week for a conference on metric geometry. I’ve listened to several nice talks already but one that stood out for me was by Yevgeny Liokumovich on the problem of cutting a sphere in half. (It had, of course, a more official title!)

Consider the sphere \(S^2\) with some Riemannian metric, scaled so that the total area is 1. Is there an upper bound to the length of a geodesic loop that divides the sphere into two disks of equal area?

It seems plausible at first that the answer might be “yes”, but in fact it is “no”. To see the counterexample, think about balloon animals: specifically a “balloon starfish” that has three thin, cylindrical arms of length \(\ell\) emanating from a central core. Continue reading