Vitali Milman apparently drew high-dimensional convex bodies as "spiky" to try to get at this intuition. So the n-dimensional cube in this case would look like

Vitali Milman apparently drew high-dimensional convex bodies as "spiky" to try to get at this intuition. So the n-dimensional cube in this case would look like a starfish, with the balls inscribed in the subdivided cubes way out in the tentacles. When you draw it this way, of course the middle ball is not contained in the cube (it is hard to reason precisely about this picture, it does not encode an obvious precise analogy).