Problems of diameters of convexes in the plane (Q2706060)

This interesting paper is devoted to the investigation of some problems on diameters of convexes in the plane. The author introduces the \(t_n\)-diameter of a compact convex set \(X\) provided that \(n\geq 2\) by the equality \(t_n(X)=\sup_{(\alpha_i)\in X^n}(\prod_{i\neq j}|\alpha_i-\alpha_j|)^{1/n(n-1)}\), which represents the diameter of \(X\). It is considered an interesting case when \(n=3\). It is well known that the length of the boundary of \(X\), \(L(X)\leq \pi t_2(X)\) and that \(\pi \) is the best constant possible. It seems natural to ask what becomes of these results if one replaces \(t_2\) by \(t_3\), hence what is the best constant \(c\) such that \(L(X)\leq c t_3(X)\). The main result is that there exists a special value of \(c\) if \(X\) has an axis and a center of symmetry.