A contribution to the Aleksandrov conservative distance problem in two dimensions

Let $E$ be a two-dimensional real normed space. In this paper we show that if the unit circle of $E$ does not contain any line segment such that the distance between its endpoints is greater than 1, then every transformation $\phi\colon E\to E$ which preserves the unit distance is automatically an affine isometry. In particular, this condition is satisfied when the norm is strictly convex.