Structure of sets with nearly maximal Favard length

Abstract

Let E Ă Bp1q Ă R 2 be an H 1 measurable set with H 1 pEq ă 8, and let L Ă R 2 be a line segment with H 1 pLq " H 1 pEq. It is not hard to see that FavpEq ď FavpLq. We prove that in the case of near equality, that is,the set E can be covered by an -Lipschitz graph, up to a set of length . The dependence between and δ is polynomial: in fact, the conclusions hold with " Cδ 1{70 for an absolute constant C ą 0.