A Problem of Fejes L. Tóth

Let $P$ be a convex $n$-gon on the Euclidean plane with edges of lengths $a_1,\dots,a_n$. Denote by $b_i$ the length of the maximal chord of $P$ parallel to $a_i$. For the quantity $\mu(P)=\sum_{i=1}^n{a_i}/{b_i}$, we prove the inequality $3\le\mu(P)\le 4$, which is the Fejes Tóth conjecture. We also give a classification of polygons with $\mu(P)=3$ or $\mu(P)=4$.