  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.
