Planets are (very likely) in orbits of stars

Abstract

The probability that a randomly and uniformly chosen point from the circumball of a tetrahedron lies outside of the inscribed ball of the tetrahedron can be bounded very sharply from below in terms of the edge lengths of the tetrahedron. One can imagine four stars in the Universe (vertices) with known mutual distances and a small (exo-) planet orbiting between them within the circumsphere. The least probability that the planet is outside of the insphere is given in terms of the distances of the stars. The least probability occurs for the regular tetrahedron and it is 0.962962. . . . Geometrically, this is a tricky corollary of (refinements of) the famous Euler inequality: circumradius is at least three times bigger than the inradius of a tetrahedron with equality for a regular tetrahedron. The Euler inequality can be extended to Euclidean sim-plices in all dimensions and to non-Euclidean planes. The most relevant cases of 3D and 4D being in accordance with the relativity theory are considered.