Are half spaces in Hadamard manifold geodesically convex?

Question

Given a Hadamard manifold $M$ (complete, simply connected and of nonpositive curvature) and two points $x,y\in M$ I want to consider the half space $H(x,y)=\{z\in M\mid d(x,z)\leq d(z,y)\}$. I wonder whether $H(x,y)$ is geodesically convex, i.e. if $\gamma\colon \mathbb R \to M$ is a geodesic with $\gamma(0)\in H(x,y)$ and $\gamma(t_0)\in H(x,y)$ for some $t_0>0$ then $\gamma(t)\in H(x,y)$ for all $t\in[0,t_0]$. I feel like this should be true as I checked the examples of the hyperbolic plane and $\mathbb R^n$. I kind of understand that the function $t\mapsto d(x,\gamma(t))$ is convex, so that the metric ball is geodesically convex. But I don't know how this helps here since we have to consider the difference of two such functions.