Let the metric space (X,d) be a CAT(0) space with the metric d. I am trying to show that the function $d_p(x)=d(p,x)$ where p is fixed point and x varies is a convex function by using the definition of convex function for metric space. A function $f:X\rightarrow R$ is said to be convex if for a geodesic $c:[0,1]\rightarrow X$ $f\circ c)$ is convex. If $s\in [0,1]$ the $f(c(s))\leq (1-s)f(c(0)+sf(c(1))$.
By CAT(0) condition $d_p(c(s))\leq \bar d(\bar{p},\bar c(s)) $. If I can show that inequality hold for comparison triangle for euclidean case I am done. ($\bar d$ is the euclidean metric in $R^2$).