A non-geodesic analogue of Reshetnyak’s majorization theorem

Abstract

Abstract For any real number κ \kappa and any integer n ≥ 4 n\ge 4 , the Cycl n ( κ ) {{\rm{Cycl}}}_{n}\left(\kappa ) condition introduced by Gromov (CAT(κ)-spaces: construction and concentration, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 280 (2001), (Geom. i Topol. 7), 100–140, 299–300) is a necessary condition for a metric space to admit an isometric embedding into a CAT ( κ ) {\rm{CAT}}\left(\kappa ) space. For geodesic metric spaces, satisfying the Cycl 4 ( κ ) {{\rm{Cycl}}}_{4}\left(\kappa ) condition is equivalent to being CAT ( κ ) {\rm{CAT}}\left(\kappa ) . In this article, we prove an analogue of Reshetnyak’s majorization theorem for (possibly non-geodesic) metric spaces that satisfy the Cycl 4 ( κ ) {{\rm{Cycl}}}_{4}\left(\kappa ) condition. It follows from our result that for general metric spaces, the Cycl 4 ( κ ) {{\rm{Cycl}}}_{4}\left(\kappa ) condition implies the Cycl n ( κ ) {{\rm{Cycl}}}_{n}\left(\kappa ) conditions for all integers n ≥ 5 n\ge 5 .