Gluing of multiple Alexandrov spaces

Perel'man's Doubling Theorem [11] says that the doubling of an Alexandrov space is also with same lower curvature bound. Petrunin [12] generalizes this theorem to two spaces being glued along their isometric boundaries. These theorems have a lot applications study bounds. As generalization these results and tool construct verify new spaces, we gluing multiple spaces. We consider Gluing Conjecture, which finite number space, if only by path isometry boundaries tangent cones are be prove Conjecture when complement (n−1,ϵ)-regular points discrete in part. In particular, implies dimension 2. This result case 2-dimensional convex sets. our proof, establish some structural theory for general gluing. Based on cone condition, 2-point over classified locally separable, as assumed Petrunin's Theorem. The near un-glued point involutional.