  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.

Metric Geometry