General quantitative Vitushkin inequality

Establish whether there is an absolute constant C such that Fav(K) ≤ Cγ(K) for every compact set K ⊂ ℂ, thereby resolving the general quantitative Vitushkin inequality.

Background

The paper discusses the quantitative form of Vitushkin’s question, relating Favard length to analytic capacity for arbitrary compact subsets of the plane. It proves superlinear comparisons for neighbourhoods of particular explicitly constructed self-similar sets, but those results do not settle the inequality for all compact sets.

The paper explicitly distinguishes its self-similar examples from the unrestricted problem and states that the general inequality remains unresolved.