Isometric embedding of Busemann surfaces into $L_1$

TL;DR

This paper proves that non-positively curved 2D surfaces, called Busemann surfaces, can be embedded into L1 space without distortion, and applies this to show certain planar graphs are also embeddable with low distortion.

Contribution

It establishes isometric embedding of Busemann surfaces into L1 and improves understanding of planar graph embeddings with non-positive curvature.

Findings

Busemann surfaces are isometrically embeddable into L1.

Planar graphs from non-positively curved complexes are embeddable into L1 with distortion less than 4.

Results simplify previous work on non-positive curvature and planar embeddings.

Abstract

In this paper, we prove that any non-positively curved 2-dimensional surface (alias, Busemann surface) is isometrically embeddable into $L_1$. As a corollary, we obtain that all planar graphs which are 1-skeletons of planar non-positively curved complexes with regular Euclidean polygons as cells are $L_1$-embeddable with distortion at most $2+\pi/2<4$. Our results significantly improve and simplify the results of the recent paper {\it A. Sidiropoulos, Non-positive curvature, and the planar embedding conjecture, FOCS 2013.}}