Isometric embedding of Busemann surfaces into $L_1$

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 paper {\it A. Sidiropoulos, Non-positive curvature, and the planar embedding conjecture, FOCS 2013.}}