Non-positive Curvature and the Planar Embedding Conjecture - Sidiropoulos, Anastasios (Annual Symposium on Foundations of Computer Science, 2013)
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.