A Big Cube and a Little Cube

On his desk, Bengt has two cubes. The small cube has the side 2 dm, and the big cube has the side 1 dm. The reason why the 1 dm cube is "big" is that Bengt claims it has more than three dimensions. Now Bengt wants to place the small cube inside the little cube. (The thickness of the walls of the cubes are negligible, so that for example if you had two identical cubes you could place one inside the other.)

1. How many dimensions must the big cube have for the little cube to fit inside it?
   
2. What if they were spheres instead of cubes?