The Euler and Grace-Danielsson inequalities for nested triangles and tetrahedra: a derivation and generalisation using quantum information theory

Abstract

We derive several results in classical Euclidean elementary geometry using the steering ellipsoid formalism from quantum mechanics. This gives a physically motivated derivation of very non-trivial geometric results, some of which are entirely new. We consider a sphere of radius $r$ contained inside another sphere of radius $R$, with the sphere centres separated by distance $d$. When does there exist a nested tetrahedron circumscribed about the smaller sphere and inscribed in the larger? We derive the Grace-Danielsson inequality $d^2 \leq (R+r)(R-3r)$ as the sole necessary and sufficient condition for the existence of a nested tetrahedron. Our method also gives the condition $d^2 \leq R(R-2r)$ for the existence of a nested triangle in the analogous 2-dimensional scenario. These results imply the Euler inequality in 2 and 3 dimensions. Furthermore, we formulate a new inequality that applies to the more general case of ellipses and ellipsoids.