The Reshetnyak majorization theorem (see 9.56) states that any closed rectifiable curve $\alpha$ in a CAT(0) length space $U$ can be majorized by a convex plane figure $F$; that is, there is short (= 1-Lipschitz) map (= majorization) $m\colon F\to U$ such that $m|_{\partial F}$ is the arc-length parametrization of $\alpha$.

Note that majorization does not decrease the curvature; that is, curvature of $\alpha$ cannot be smaller than curvature of $\partial F$ at the corresponding point. Therefore, the inequality $$ \ell \ge 2\cdot \pi\cdot \varepsilon^{-1}$$ holds in CAT(0) spaces as well.

The Reshetnyak majorization theorem (see 9.56) states that any closed rectifiable curve $\alpha$ in a CAT(0) length space $U$ can be majorized by a convex plane figure $F$; that is, there is short (= 1-Lipschitz) map (= majorization) $m\colon F\to U$ such that $m|_{\partial F}$ is the arc-length parametrization of $\alpha$.

Note that majorization does not decrease the curvature; that is, curvature of $\alpha$ cannot be smaller than curvature of $\partial F$ at the corresponding point. Therefore, the inequality $$ \ell \ge 2\cdot \pi\cdot \varepsilon^{-1}$$ holds in CAT(0) spaces as well.