Non-positive curvature, and the planar embedding conjecture

TL;DR

This paper proves that all planar metrics with non-positive curvature can be embedded into L_1 space with constant distortion, advancing the longstanding planar embedding conjecture.

Contribution

It establishes the conjecture for non-positively curved planar metrics, a significant step beyond previous restricted cases.

Findings

All non-positively curved planar metrics admit constant-distortion L_1 embeddings.

The result confirms the conjecture for a broad class of planar metrics.

Progress towards resolving the general planar embedding conjecture.

Abstract

The planar embedding conjecture asserts that any planar metric admits an embedding into L_1 with constant distortion. This is a well-known open problem with important algorithmic implications, and has received a lot of attention over the past two decades. Despite significant efforts, it has been verified only for some very restricted cases, while the general problem remains elusive. In this paper we make progress towards resolving this conjecture. We show that every planar metric of non-positive curvature admits a constant-distortion embedding into L_1. This confirms the planar embedding conjecture for the case of non-positively curved metrics.