Structure of sets with nearly maximal Favard length

Abstract

Let E⊂B(1)⊂R2 be an H1 measurable set with H1(E)<∞, and let L⊂R2 be a line segment with H1(L)=H1(E). It is not hard to see that Fav(E)≤Fav(L). We prove that in the case of near equality, that is, Fav(E)≥Fav(L)−δ, the set E can be covered by an ϵ-Lipschitz graph, up to a set of length ϵ. The dependence between ϵ and δ is polynomial: in fact, the conclusions hold with ϵ=Cδ1∕70 for an absolute constant C>0.