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 two-dimensional scenario. These results imply the Euler inequality in two and three dimensions. Furthermore, we formulate a new inequality that applies to the more general case of ellipses and ellipsoids.
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Milne, A. The Euler and Grace-Danielsson inequalities for nested triangles and tetrahedra: a derivation and generalisation using quantum information theory. J. Geom. 106, 455–463 (2015). https://doi.org/10.1007/s00022-014-0257-8
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DOI: https://doi.org/10.1007/s00022-014-0257-8