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.



Generalized Sphere Kissing Problem
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](http://arxiv.org/abs/1510.00407))


