This question is cross-posted in MO and MSE <https://math.stackexchange.com/questions/2276064/about-pairwise-distances-of-some-points-in-a-riemannian-manifold-m-of-rm-se>



> 
> Assume that there are points $p\_i,\ 1\leq i\leq m$ in a Riemannian
>  manifold $(M,d)$ s.t.
> 
> 
> (1) all sectional curvatures are equal to or greater than $1$ and
> 
> 
> (2) $$ d(p\_i,p\_j)> \frac{\pi}{2},\ i\neq j $$
>  Then there is a set of points $\eta\_{ij},\ i\neq j$ s.t. $$
> d(p\_i,\eta\_{ij}) < \frac{\pi}{2} <d(p\_j,\eta\_{ij}) $$ and $$
> d(p\_l,\eta\_{ij} )=\frac{\pi}{2} $$ for all $l$ not in $\{ i,j\}$.
> 
> 
> 


How do we prove this ? Thank you in advance.


**[Add]**


(a) $\angle p\_ip\_jp\_k > \frac{\pi}{2}$


(b) Note that if there exists such $\eta\_{ij}$, then $|\{ \eta\_{ij}
\}|=m(m-1)$.


(c) $m=3$ Case is solved : Consider a geodesic triangle
$[p\_1p\_2p\_3]$. Define $\eta\_{31}\in [p\_2p\_3]$ s.t. $d(p\_2,\eta\_{31}
)= \frac{\pi}{2}$. By applying Toponogov theorem to a hinge at
$p\_2$, $d(p\_1,\eta\_{31}) > \frac{\pi}{2}$.


(d) My difficulty is to find $\eta\_{m(m-1)}$


By an induction, there is $q$ around $p\_m$, which may help to find
$\eta\_{m(m-1)}$, s.t.
$$l\_i:=d(q,p\_i)=\frac{\pi}{2},\ 1\leq i\leq m-3,\ L:=d(q,p\_{m-3}) > \frac{\pi}{2} $$


Hence there is a point that increases $l\_i$ and diminishes $L$



About pairwise distances of some points in a Riemannian manifold $M$ of ${\rm sec}\ M\geq 1$
We can assume that the set $\{p\_1,\dots,p\_n\}$ is maximal.
Note that the set 
$$K\_{n,n-1}=\{\,x\in M\mid |x-p\_i|\ge\tfrac\pi2\ \text{for}\ i<n-1\,\}$$
is convex and it contains a point $z$ such that $|p\_i-z|=\tfrac\pi2$ for any $i<n$.


Since $\{p\_1,\dots,p\_n\}$ is maximal, $|p\_n-z|\le \tfrac\pi2$.
By comparison, we get 
$$\measuredangle[z\,^{p\_i}\_{p\_j}]>\tfrac\pi2$$
for all $i\ne j$.


Flow $z$ in the gradient flow for $\mathrm{dist}\_{p\_{n-1}}$ in $K\_{n,n-1}$ for short time. We get a point $z'$ which will satisfy all your conditions for $\eta\_{n,n-1}$.


