Banach-Minkowski Planes

Abstract

In the former chapters we calculated or estimated the Steiner ratio for many metric spaces. We have seen that this quantity of a space lies between 0.5 and 1 and, moreover, these bounds are the best possible ones, even for finite-dimensional Banach spaces. In the present chapter we will see that the last fact is not true in two-dimensional Banach spaces. That means we will create better upper and lower bounds for the Steiner ratio of Banach-Minkowski planes.We find in theorem 7.4.1 that for any unit ball B in the plane the following is true:

$${m_2}\left( B \right)\frac{{\sqrt {13} - 1}}{3} = 0.8685...$$

A better bound is still unknown.