I am looking for a proof of the following statement:
Let $X$ be a complete Busemann space. For any point $x\in X$ and any nonempty closed convex set $A\subseteq X$, there is a unique $a\in A$ such that $d(x,a) = d(x,A)$.
By a Busemann space, I mean a geodesic metric space such that for any two affinely parametrized geodesics, $\gamma_0$ and $\gamma_1$, the map $(s,t)\mapsto d(\gamma_0(s),\gamma_1(t))$ is convex. In Papadopoulos' Metric Spaces, Convexity and Nonpositive Curvature, the above statement is given as Proposition 8.4.8. However, I found the proof rather unconvincing as it addresses uniqueness but not existence (existence is not clarified earlier in the text). In Bridson and Haefliger's Metric Spaces of Non-positive Curvature there is a nice proof for CAT(0) spaces, but it doesn't seem to directly generalize to the weaker curvature condition of Busemann spaces.