Isometric Embedding of Busemann Surfaces into $L_1$

International audienceIn this paper, 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. Our results significantly improve and simplify the results of the recent paper by A. Sidiropoulos (Non-positive curvature and the planar embedding conjecture, FOCS (2013))