Isometric Embeddings of Polyhedra into Euclidean Space

TL;DR

This paper proves that Euclidean polyhedra can be approximated by isometric embeddings in Euclidean space, extending classical theorems to piecewise linear and other polyhedral geometries.

Contribution

It establishes new approximation results for isometric embeddings of Euclidean, spherical, and hyperbolic polyhedra into Euclidean space, generalizing Nash-Kuiper theorem.

Findings

Any 1-Lipschitz map from an n-dimensional Euclidean polyhedron into n is close to a PL isometric embedding.

Any 1-Lipschitz map into 2n+1 can be approximated by a continuous isometric embedding.

Results extend to spherical and hyperbolic polyhedra using Nash-Kuiper theorem.

Abstract

In this paper we consider piecewise linear (pl) isometric embeddings of Euclidean polyhedra into Euclidean space. A Euclidean polyhedron is just a metric space $\mathcal{P}$ which admits a triangulation $\mathcal{T}$ such that each $n$-dimensional simplex of $\mathcal{T}$ is affinely isometric to a simplex in $\mathbb{E}^n$. We prove that any 1-Lipschitz map from an $n$-dimensional Euclidean polyhedron $\mathcal{P}$ into $\mathbb{E}^{3n}$ is $\epsilon$-close to a pl isometric embedding for any $\epsilon > 0$. If we remove the condition that the map be pl then any 1-Lipschitz map into $\mathbb{E}^{2n + 1}$ can be approximated by a (continuous) isometric embedding. These results are extended to isometric embedding theorems of spherical and hyperbolic polyhedra into Euclidean space by the use of the Nash-Kuiper $C^1$ isometric embedding theorem. Finally, we discuss how these results extend to various other types of polyhedra.