| Summary: | For any real number κ\kappa and any integer n≥4n\ge 4, the Cycln(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 Cycl4(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 Cycl4(κ){{\rm{Cycl}}}_{4}\left(\kappa ) condition. It follows from our result that for general metric spaces, the Cycl4(κ){{\rm{Cycl}}}_{4}\left(\kappa ) condition implies the Cycln(κ){{\rm{Cycl}}}_{n}\left(\kappa ) conditions for all integers n≥5n\ge 5.
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