Quadratic Crofton and sets that see themselves as little as possible

Abstract

Let \(\Omega \subset \mathbb {R}^2\) and let \(\mathcal {L} \subset \Omega \) be a one-dimensional set with finite length \(L =|\mathcal {L}|\) . We are interested in minimizers of an energy functional that measures the size of a set projected onto itself in all directions: we are thus asking for sets that see themselves as little as possible (suitably interpreted). Obvious minimizers of the functional are subsets of a straight line but this is only possible for \(L \le \text{ diam }(\Omega )\) . The problem has an equivalent formulation: the expected number of intersections between a random line and \(\mathcal {L}\) depends only on the length of \(\mathcal {L}\) (Crofton’s formula). We are interested in sets \(\mathcal {L}\) that minimize the variance of the expected number of intersections. We solve the problem for convex \(\Omega \) and slightly less than half of all values of L: there, a minimizing set is the union of copies of the boundary and a line segment.