Continuality of set of biLipschitz classes in Euclidean space

TL;DR

This paper proves that in Euclidean space of dimension at least 2, the set of biLipschitz classes is uncountably infinite, specifically having the cardinality of the continuum.

Contribution

It establishes that the collection of biLipschitz classes in Euclidean spaces of dimension two or higher is uncountably infinite, expanding understanding of geometric classification.

Findings

The set of biLipschitz classes in ^d has continuum cardinality for all d  2.

BiLipschitz classes are highly diverse in Euclidean spaces.

The result applies to all Euclidean spaces of dimension at least 2.

Abstract

The main result of this paper is Theorem. For every integer $d\geqslant 2$ the set of biLipschitz classes in $\mathbb{E}^d$ has cardinality continuum.