A finite quotient of join in Alexandrov geometry

Abstract:

Given two $n_i$-dimensional Alexandrov spaces $X_i$ of curvature $\ge 1$, the join of $X_1$ and $X_2$ is an $(n_1+n_2+1)$-dimensional Alexandrov space $X$ of curvature $\ge 1$, which contains $X_i$ as convex subsets such that their points are $\frac \pi 2$ apart. If a group acts isometrically on a join that preserves $X_i$, then the orbit space is called a quotient of join. We show that an $n$-dimensional Alexandrov space $X$ with curvature $\ge 1$ is isometric to a finite quotient of join, if $X$ contains two compact convex subsets $X_i$ without boundary such that $X_1$ and $X_2$ are at least $\frac \pi 2$ apart and $\dim (X_1)+\dim (X_2)=n-1$.