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Let $X$ be a CAT(0) space and suppose $a,b,c,d\in X$ satisfy $$ \max\{\angle_a(b,c),\angle_a(b,d)\}<\frac{\pi}{2}. $$ Let $\gamma:[0,\ell]\to X$ be the geodesic with $\gamma(0)=c$ and $\gamma(\ell)=d$.

Is the function $$ t \mapsto \angle_a(b,\gamma(t)) $$ convex? If not, is it at least the case that for all $t\in[0,\ell]$, $$ \angle_a(b,\gamma(t)) \leq \max\{\angle_a(b,c),\angle_a(b,d)\}? $$

Let $X$ be a CAT(0) space and suppose $a,b,c,d\in X$ satisfy $$ \max\{\angle_a(b,c),\angle_a(b,d)\}<\frac{\pi}{2}. $$ Let $\gamma:[0,\ell]\to X$ be the geodesic with $\gamma(0)=c$ and $\gamma(\ell)=d$.

Is it true that for all $t\in[0,\ell]$, $$ \angle_a(b,\gamma(t)) \leq \max\{\angle_a(b,c),\angle_a(b,d)\}? $$

Let $X$ be a CAT(0) space, let $a,b,c,d\in X$ and let $\gamma:[0,\ell]\to X$ be the geodesic with $\gamma(0)=c$ and $\gamma(\ell)=d$. Is the function $$ t \mapsto \angle_a(b,\gamma(t)) $$ convex? If not, is it at least the case that for all $t\in[0,\ell]$, $$ \angle_a(b,\gamma(t)) \leq \max\{\angle_a(b,c),\angle_a(b,d)\}? $$

Let $X$ be a CAT(0) space and suppose $a,b,c,d\in X$ satisfy $$ \max\{\angle_a(b,c),\angle_a(b,d)\}<\frac{\pi}{2}. $$ Let $\gamma:[0,\ell]\to X$ be the geodesic with $\gamma(0)=c$ and $\gamma(\ell)=d$. 

Is the function $$ t \mapsto \angle_a(b,\gamma(t)) $$ convex? If not, is it at least the case that for all $t\in[0,\ell]$, $$ \angle_a(b,\gamma(t)) \leq \max\{\angle_a(b,c),\angle_a(b,d)\}? $$