A problem of Fejes L. Tóth (Q1876365)

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 \(\mu(P)=\sum_{i=1}^{n} a_i/b_i\), the authors prove the inequality \(3\leq \mu(P)\leq 4\) which was conjectured by László Fejes Tóth (whose name is given in a strange form in the title of the paper under review). The authors also find all polygons with \(\mu(P)=3\) or \(\mu(P)=4\). This is an English translation of the authors' article [Mat. Tr. 5, No. 1, 102--113 (2002; Zbl 1015.52005)].