Consider when the top layer is rotated 45 degrees relative to the layer below. The corner is now about 0.707 units from the centre of the face, but the layer below only extends 0.5 units out from the centre.

If the corner pieces were smaller than about 0,207 units it would become disconnected from the cube as a whole.

So, any cube larger than (I believe) the 5 by 5 either needs to have bulging faces, or have the edge/corner pieces larger than the others.