A non-geodesic analogue of Reshetnyak's majorization theorem

For any real number ? and any integer n = 4, the Cycl(n)(?) condition introduced by Gromov (CAT (?)-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(?) space. For geodesic metric spaces, satisfying the Cycl(4)(?) condition is equivalent to being CAT(?). In this article, we prove an analogue of Reshetnyak's majorization theorem for (possibly non-geodesic) metric spaces that satisfy the Cycl(4)(?) condition. It follows from our result that for general metric spaces, the Cycl(4)(?) condition implies the Cycl(n)(?) conditions for all inte-gers n=5.