On $\frac\pi2$-separated subsets of Alexandrov spaces with curvature $\geq1$

Abstract

Let $M$ be an $n$-dimensional Alexandrov space with curvature $\geq 1$, and let $\{q_1,\cdots,q_k\}$ be any $\frac\pi2$-separated subset in $M$ (i.e. the distance $|q_iq_j|\geq\frac{\pi}{2}$ for any $i\neq j$). Under the additional conditions "$|q_iq_j|<\pi$" and "the diameter $\diam(M)\leq \frac\pi2$", we respectively give the upper bound of $k$ (which depends only on $n$), and we classify the (topological or geometric) structure of $M$ when $k$ attains the upper bound.