Isometry Classes of Planes in $(\mathbb{R}^3,d_{\infty})$

TL;DR

This paper characterizes geodesics in the max-norm space and uses this to classify all planes up to isometry in three-dimensional max-norm space, advancing understanding of geometric structures in this metric.

Contribution

It provides a complete classification of planes up to isometry in ^3 with the max-norm by analyzing geodesic structures.

Findings

Geodesics in ^n are explicitly determined.

Planes in ^3 are classified up to isometry based on geodesic properties.

Abstract

We determine geodesics in $\mathbb{R}_{\infty}^n$ (i.e. $(\mathbb{R}^n, d_{\infty})$) and by using this, classify planes up to isometry in $\mathbb{R}_{\infty}^3$.