  In this paper we determine new upper bounds for the maximal density of
translative packings of superballs in three dimensions (unit balls for the
$l^p_3$-norm) and of Platonic and Archimedean solids having tetrahedral
symmetry. Thereby, we improve Zong's recent upper bound for the maximal density
of translative packings of regular tetrahedra from $0.3840\ldots$ to
$0.3745\ldots$, getting closer to the best known lower bound of $0.3673\ldots$
  We apply the linear programming bound of Cohn and Elkies which originally was
designed for the classical problem of densest packings of round spheres. The
proofs of our new upper bounds are computational and rigorous. Our main
technical contribution is the use of invariant theory of pseudo-reflection
groups in polynomial optimization.
