Structure of sets with nearly maximal Favard length

Let $E\subset B(1)\subset {ℝ}^2$ be an ${\mathcal{\mathscr{H}}}^1$ measurable set with ${\mathcal{\mathscr{H}}}^1(E)<\infty$, and let $L\subset {ℝ}^2$ be a line segment with ${\mathcal{\mathscr{H}}}^1(L)={\mathcal{\mathscr{H}}}^1(E)$. It is not hard to see that $\mathrm{Fav}<!--FUNCTION\; APPLICATION-->(E)\le \mathrm{Fav}<!--FUNCTION\; APPLICATION-->(L)$. We prove that in the case of near equality, that is,

$$\mathrm{Fav}<!--FUNCTION\; APPLICATION-->(E)\ge \mathrm{Fav}<!--FUNCTION\; APPLICATION-->(L)-\delta ,$$

the set $E$ can be covered by an $𝜖$-Lipschitz graph, up to a set of length $𝜖$. The dependence between $𝜖$ and $\delta$ is polynomial: in fact, the conclusions hold with $𝜖=C{\delta }^{1∕70}$ for an absolute constant $C>0$.