Grandcolas, Michel Problems of diameters of convexes in the plane. (English) Zbl 0973.12001 Int. J. Differ. Equ. Appl. 1A, No. 4, 409-413 (2000). 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. Reviewer: Dimitar Kolev (Sofia) MSC: 12D10 Polynomials in real and complex fields: location of zeros (algebraic theorems) 52A10 Convex sets in \(2\) dimensions (including convex curves) 11R09 Polynomials (irreducibility, etc.) Keywords:weighted diameter; isoperimetric inequalities PDFBibTeX XMLCite \textit{M. Grandcolas}, Int. J. Differ. Equ. Appl. 1A, No. 4, 409--413 (2000; Zbl 0973.12001)