Quadratic Crofton and sets that see themselves as little as possible

TL;DR

This paper investigates sets in the plane that minimize their self-projection size by analyzing the variance of intersections with random lines, identifying optimal configurations for convex domains and specific lengths.

Contribution

It introduces a novel variational problem based on Crofton's formula and characterizes minimizers as unions of boundary segments and lines for certain lengths.

Findings

Minimizers are unions of boundary segments and lines for convex domains and specific lengths.

The problem reduces to minimizing the variance of line intersections, not just the expected value.

Explicit solutions are provided for convex sets and lengths less than half the domain diameter.

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 $L \leq \mbox{diam}(\Omega)$. 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.