A note on Brehm's extension theorem

TL;DR

This paper verifies that Brehm's extension theorem can be constructively applied in two dimensions when the finite subset and the non-expansive map involve rational coordinates, enabling explicit extensions.

Contribution

It demonstrates the constructiveness of Brehm's extension theorem in two dimensions for rational-coordinate points and maps.

Findings

Constructive proof for rational points in 2D

Extension of non-expansive maps to piecewise-linear maps

Applicable to finite rational-coordinate sets

Abstract

Brehm's extension theorem states that a non-expansive map on a finite subset of a Euclidean space can be extended to a piecewise-linear map on the entire space. In this note, it is verified that the proof of the theorem is constructive provided that the finite subset consists of points with rational coordinates. Additionally, the initial non-expansive map needs to send points with rational coordinates to points with rational coordinates. The two-dimensional case is considered.