Isometric embedding of Busemann surfaces into L1

Abstract. In this note, we prove that any non-positively curved 2-dimensional surface (alias, Busemann surface) is isometrically embeddable into L1. 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 L1-embeddable with distortion at most 2 + pi/2 < 4. Our results significantly improve and simplify the results of the recent paper A. Sidiropoulos, Non-positive curvature, and the planar embedding conjecture, FOCS 2013. 1. Avant-propos Isometric and low distortion embeddings of finite and infinite metric spaces into Lp-spaces is one of the main subjects in the theory of metric spaces. Work in this area was initiated by Cayley in 1841 and continued in the first half of the 20th century by Fréchet, Menger, Schoenberg, and Blumenthal. Since these days it is known that all metric spaces isometri-cally embed into L∞. Metric spaces isometrically embeddable into L2 were characterized by Menger and Schoenberg. Even if embeddability into L1 can be defined in several equivalent ways and a few necessary conditions for L1-embedding are known, metric spaces isometrically embeddable into L1 cannot be characterized in an efficient way, because deciding whether a finite metric space is L1-embeddable is NP-complete. On the other hand, many classe