On Banach–Mazur distance between planar convex bodies

Upper estimates of the diameter and the radius of the family of planar convex bodies with respect to the Banach–Mazur distance are obtained. Namely, it is shown that the diameter does not exceed 19-734≈2.614\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tfrac{19-\sqrt{73}}{4}\approx 2.614$$\end{document}, which improves the previously known bound of 3, and that the radius does not exceed 11770≈1.671\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{117}{70}\approx 1.671$$\end{document}.