Quadratic Crofton and sets that see themselves as little as possible

Let Ω⊂ℝ^2 and let ℒ⊂Ω be a one-dimensional set with finite length L =|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 ≤ diam (Ω ) . The problem has an equivalent formulation: the expected number of intersections between a random line and ℒ depends only on the length of ℒ (Crofton’s formula). We are interested in sets ℒ that minimize the variance of the expected number of intersections. We solve the problem for convex Ω 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.