A non-geodesic analogue of Reshetnyak's majorization theorem

For any real number $\kappa$ and any integer $n\ge 4$, the $$\mathrm{C}\mathrm{y}\mathrm{c}\mathrm{l}$_n (\kappa )$ condition introduced by Gromov (2001) is a necessary condition for a metric space to admit an isometric embedding into a $\mathrm{C}\mathrm{A}\mathrm{T}(\kappa )$ space. It is known that for geodesic metric spaces, the $$\mathrm{C}\mathrm{y}\mathrm{c}\mathrm{l}$_4 (\kappa )$conditionisequivalenttobeing$$\mathrm{C}\mathrm{A}\mathrm{T}$(\kappa )$. In this paper, we prove an analogue of Reshetnyak’s majorization theorem for (possibly non-geodesic) metric spaces that satisfy the ${\mathrm{C}\mathrm{y}\mathrm{c}\mathrm{l}}_4(\kappa )$ condition. It follows from our result that for general metric spaces, the ${\mathrm{C}\mathrm{y}\mathrm{c}\mathrm{l}}_4(\kappa )$ condition implies the ${\mathrm{C}\mathrm{y}\mathrm{c}\mathrm{l}}_n(\kappa )$ conditions for all integers $n\ge 5$, although Gromov stated that this implication is apparently not true.