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 are 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 $\eta_{31}\in [p_2p_2]$ 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 $L$