Structure of sets with nearly maximal Favard length

TL;DR

This paper characterizes the structure of sets in the plane with Favard length close to maximal, showing they are nearly Lipschitz graphs with a polynomial relation between the approximation parameters.

Contribution

It establishes that sets with nearly maximal Favard length are structurally close to Lipschitz graphs, quantifying the approximation with a polynomial dependence.

Findings

Sets with near-maximal Favard length are covered by Lipschitz graphs.

The relation between the approximation parameters is polynomial.

Provides a structural stability result for Favard length near extremizers.

Abstract

Let $E \subset B(1) \subset \mathbb R^{2}$ be an $\mathcal{H}^{1}$ measurable set with $\mathcal{H}^{1}(E) < \infty$, and let $L \subset \mathbb R^{2}$ be a line segment with $\mathcal{H}^{1}(L) = \mathcal{H}^{1}(E)$. It is not hard to see that $\mathrm{Fav}(E) \leq \mathrm{Fav}(L)$. We prove that in the case of near equality, that is, $$ \mathrm{Fav}(E) \geq \mathrm{Fav}(L) - \delta, $$ the set $E$ can be covered by an $\epsilon$-Lipschitz graph, up to a set of length $\epsilon$. The dependence between $\epsilon$ and $\delta$ is polynomial: in fact, the conclusions hold with $\epsilon = C\delta^{1/70}$ for an absolute constant $C > 0$.