# Monotonicity of metric projection onto a closed convex set

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?