Upper Bounds for Non-Congruent Sphere Packings

We prove upper bounds on the average kissing number $k(\mathcal{P})$ and contact number $C(\mathcal{P})$ of an arbitrary finite non-congruent sphere packing $\mathcal{P}$, and prove an upper bound on the packing density $δ(\mathcal{P})$ of an arbitrary infinite non-congruent sphere packing $\mathcal{P}$.

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