Hide and seek with curves in the plane
Suppose we fix a convex domain in the plane and fix a prescribed length. What curve can we fit inside this domain so that the curve sees itself as little as possible; that is, so that it minimizes an energy functional that projects it onto itself in all directions? Using a tool from integral geometry called Crofton's formula, we can reformulate this in terms of minimizing the variance of the number of intersections of a random line with our curve. This is based on a very recent paper of Stefan Steinerberger, and I'll discuss some of the manifold open problems and conjectures surrounding this.