Maybe this question was already asked and forgive me if I can't formulate it well.

Lets assume we have a n-dimensional hypercube. If we want the smallest set of vertices such that the cube is inside the convex cone of the vertices we can take all the vertices on the main coordinates - (1,0,0,..,0),(0,1,0,..,0) etc.

My question is what is the maximum number of vertices we can take, such that 1. No vertex is in the interior of the convex cone generated by these edges. 2. The cube is not contained in the convex cone of the vertices.