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I am currently researching discrete geometry and I am in need of an upper bound on a generalized kissing number in 3-dimensions dependent upon a parameter $\eta$ which is the radii of spheres touching a central unit ball.

That is, center the unit ball $B \subset \mathbb{R}^3$ at the origin and define $k_{s}(\eta)$ to be the maximum number of balls of radius $\eta$ which can kiss $B$ and form a solid packing (non-overlapping interiors). How can you upper bound $k_{s}(\eta)$ for $\eta >0$? A remarkable case of note is $k_{s}(1) = 12$ which corresponds to the solution of the Newton-Gregory problem.

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    $\begingroup$ How sharp a bound do you need? Comparing volumes gives you one, but that is horribly far from the actual number. Might be enough for certain theoretical purposes though, depending on what you're doing. $\endgroup$
    – Achim Krause
    Commented Mar 19, 2014 at 20:36
  • $\begingroup$ @AchimKrause: I am attempting to improve an upper bound regarding non-congruent sphere packings and need a bound which is as tight as possible; I'm not sure how I close need yet. I'm not sure what technique you intend to employ by comparing volumes. $\endgroup$
    – Samuel Reid
    Commented Mar 19, 2014 at 20:37
  • $\begingroup$ Well, projecting the balls kissing B onto the surface of B gives a disjoint circles on the unit 2-sphere. By computing their area and comparing it to the total area, you obtain an upper bound. For example, in the $\eta=1$-case, this gives 16, which I think is way too bad for your purposes. $\endgroup$
    – Achim Krause
    Commented Mar 19, 2014 at 20:53
  • $\begingroup$ @BenoîtKloeckner: I need bounds for large $\eta$ as well. I'm not sure I see the equivalence, would you perhaps be able to explain as the answer to this question? $\endgroup$
    – Samuel Reid
    Commented Mar 19, 2014 at 21:55
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    $\begingroup$ @SamuelReid: my point is that this question being a duplicate, it should be discussed at the question mentionned above. $\endgroup$
    – Benoît Kloeckner
    Commented Mar 20, 2014 at 14:14

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I happened to write a paper which eventually answered my problem, although it happens to be hid inside the $A^{\text{LP}}(3,\theta)$ term in the bound of Theorem 1 of my paper which is under review at the Journal of Geometry:

"Upper Bounds for Non-Congruent Sphere Packings" (arXiv link)

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