Upper bounds for packings of spheres of several radii. de Laat, D.; Filho, F. M. d. O.; and Vallentin, F. Forum of Mathematics, Sigma, September, 2014. arXiv: 1206.2608
Upper bounds for packings of spheres of several radii [link]Paper  doi  abstract   bibtex   
We give theorems that can be used to upper bound the densities of packings of different spherical caps in the unit sphere and of translates of different convex bodies in Euclidean space. These theorems extend the linear programming bounds for packings of spherical caps and of convex bodies through the use of semidefinite programming. We perform explicit computations, obtaining new bounds for packings of spherical caps of two different sizes and for binary sphere packings. We also slightly improve bounds for the classical problem of packing identical spheres.
@article{de_laat_upper_2014,
	title = {Upper bounds for packings of spheres of several radii},
	volume = {2},
	issn = {2050-5094},
	url = {http://arxiv.org/abs/1206.2608},
	doi = {10.1017/fms.2014.24},
	abstract = {We give theorems that can be used to upper bound the densities of packings of different spherical caps in the unit sphere and of translates of different convex bodies in Euclidean space. These theorems extend the linear programming bounds for packings of spherical caps and of convex bodies through the use of semidefinite programming. We perform explicit computations, obtaining new bounds for packings of spherical caps of two different sizes and for binary sphere packings. We also slightly improve bounds for the classical problem of packing identical spheres.},
	urldate = {2016-01-30TZ},
	journal = {Forum of Mathematics, Sigma},
	author = {de Laat, David and Filho, Fernando Mario de Oliveira and Vallentin, Frank},
	month = sep,
	year = {2014},
	note = {arXiv: 1206.2608},
	keywords = {52C17, 90C22, Mathematics - Metric Geometry, Mathematics - Optimization and Control}
}
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