A non-geodesic analogue of Reshetnyak's majorization theorem

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