Maximizing Volume Ratios for Shadow Covering by Tetrahedra

Define a body A to be able to hide behind a body B if the orthogonal projection of B contains a translation of the corresponding orthogonal projection of A in every direction. In two dimensions, it is easy to observe that there exist two objects such that one can hide behind another and have a larger area than the other. It was recently shown that similar examples exist in higher dimensions as well. However, the highest possible volume ratio for such bodies is still undetermined. We investigated two three-dimensional examples, one involving a tetrahedron and a ball and the other involving a tetrahedron and an inverted tetrahedron. We calculate the highest volume ratio known up to this date, 1.16, which is generated by our second example.