  Upper estimates of the diameter and the radius of the family of all planar
convex bodies with respect to the Banach-Mazur distance are obtained. Namely,
it is shown that the diameter does not exceed $\tfrac{19-\sqrt{73}}4\approx
2.614$, which improves the previously known bound of $3$, and that the radius
does not exceed $\frac{117}{70}\approx 1.671$.
