Let $X\subset\mathbb{R}^n$ be some connected and bounded $n$-dimensional manifold, e.g. a space homeomorphic to an open/closed ball with possibly some parts of it being removed.
I am interested in finding the minimum number of cones, whose vertices lie in $X$, and whose union contains $X$, or rather some bounds on this number.
One hope is that the rank of something like singular/simplicial homology can provide some, I presume, lower bounded to this. Or maybe covering by cones would allow to define another homology theory.
I am quite unfamiliar with the literature that are possibly related to this, and I would appreciate any comment or advise on this.
