  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 $C > 0$.
