# Grace–Danielsson Inequality

When can you fit a tetrahedron between two nested spheres? Suppose the radius of the large sphere is $R$ and the radius of the small one is $r$. Suppose the distance between their centers is $d$. Then you can fit a tetrahedron between these spheres if and only if the Grace–Danielsson inequality $d^2 \le (R + r)(R – 3r)$ holds. This was independently proved by Grace in 1917 and Danielsson in 1949. But Antony Milne has found a new proof of this inequality using quantum information theory!