Let $(X,d)$ be a complete $CAT(0)$ space (also known as a Hadamard space). Let $x\in H$ and $\gamma:\mathbb{R}\to H$ be some geodesic line containing $x$. For $y,z\in H$ (wlog assume $x\neq y\neq z$) let $[y,z]$ denote the geodesic segment connecting $z$ with $y$. For $\alpha\in [0,1]$ define $z_{\alpha}:=(1-\alpha)y+\alpha z$ to be the point on $[y,z]$ such that $d(y,z_{\alpha})=\alpha d(y,z)$. Let $f(\alpha):=d(x,P_{\gamma}z_{\alpha})$ where $P_{\gamma}$ is the metric projection onto $\gamma$. From the estimate $$|d(x,P_{\gamma}z_{\alpha})-d(x,P_{\gamma}z_{\beta})|\leqslant d(P_{\gamma}z_{\alpha},P_{\gamma}z_{\beta})\leqslant d(z_{\alpha},z_{\beta})=|\alpha-\beta|d(y,z)$$ the function $f(\alpha)$ is continuous in $\alpha$ (the second inequality follows since $P_{C}$ is nonexpansive whenever $C$ is closed convex set). Is $f(\alpha)$ monotone?


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