A contribution to the Aleksandrov conservative distance problem in two dimensions

TL;DR

This paper proves that in two-dimensional normed spaces with certain convexity conditions, all unit distance preserving transformations are affine isometries, extending understanding of distance-preserving maps in geometric analysis.

Contribution

It establishes that in two-dimensional normed spaces without long segments on the unit circle, all unit distance preservers are affine isometries, generalizing previous results to broader conditions.

Findings

Unit circle without long segments implies all unit distance preservers are affine isometries.

Strict convexity of the norm ensures the condition is satisfied.

The result extends the Aleksandrov conservative distance problem to new classes of spaces.

Abstract

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.