Minimal atlases of manifolds

Given a connected compact n-manifold M, a natural invariant of M is the minimal number of balls which are needed to cover M. If the intersection of any number of balls has again balls as connected components, we get the notion of ball-intersection atlas of M. We prove that each minimal ball-intersection atlas of a connected piecewise-linear n-manifold M has exactly n balls if the boundary of M is non-void