In this paper we determine new upper bounds for the maximal density oftranslative packings of superballs in three dimensions (unit balls for the$l^p_3$-norm) and of Platonic and Archimedean solids having tetrahedralsymmetry. Thereby, we improve Zong's recent upper bound for the maximal densityof 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 wasdesigned for the classical problem of densest packings of round spheres. Theproofs of our new upper bounds are computational and rigorous. Our maintechnical contribution is the use of invariant theory of pseudo-reflectiongroups in polynomial optimization.
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