A FINITE QUOTIENT OF JOIN IN ALEXANDROV GEOMETRY

Given two n(i),dimensional Alexandrov spaces X-i of curvature >= 1, the join of X-1 and X-2 is an (n(1) + n(2) + 1)-dimensional Alexandrov space X of curvature >= 1, which contains X-i as convex subsets such that their points are 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 >= 1 is isometric to a finite quotient of join, if X contains two compact convex subsets X, without boundary such that X-1 and X-2 are at least pi/2 apart and dim(X-i) + dim(X-2) = n - 1.

Automated summary

For an n-dimensional Alexandrov space X with curvature >= 1, X can be isometric to a finite quotient of join if it contains two compact convex subsets X, without boundary, such that their dimensions sum up to n-1 and they are at least pi/2 apart.