A non-geodesic analogue of Reshetnyak's majorization theorem

Abstract

For any real number $\kappa$ and any integer $n\ge 4$, 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 $n\ge 5$, although Gromov stated that this implication is apparently not true.

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