If we have a unit circle within a square s.t. the square touches the circle in 4 places then the biggest gap we can find is just √2 - 1. Doing a similar thing with a sphere in a cube we get √3 - 1 I've heard the n-dimensional analogue is √n - 1. Which is crazy as it means the gap is bigger than the radius of the sphere! (When n is greater than 4). Anyway, how is such a thing proven?