User kore min - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T00:35:14Z http://mathoverflow.net/feeds/user/1054 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/2137/a-round-lattice-with-low-kissing-number A "round" lattice with low kissing number? Kore Min 2009-10-23T18:00:22Z 2011-02-14T22:50:03Z <p>Historically, the lattices with high density were studied intensively, e.g. E_8 lattice or Leech Lattice. However, there are situations that lattices with low kissing number are required. Specifically, I am wondering if there is any infinite family of n-dimension lattices which the kissing number is bounded by poly(n), and still relatively dense?</p> http://mathoverflow.net/questions/2433/making-an-l-2-distance-out-of-l-1-distance/2434#2434 Answer by Kore Min for Making an l_2 distance out of l_1 distance Kore Min 2009-10-25T06:33:11Z 2009-10-25T06:33:11Z <p>I guess you are thinking something about metric embedding. Any metric can be embedding into l_2 with the distortion bounded by O(log n), which is tight in general.</p> <p>See this lecture notes for more information: <a href="http://www-math.mit.edu/~goemans/18409.html" rel="nofollow">http://www-math.mit.edu/~goemans/18409.html</a></p> <p>Also, the Kashinâ€™s decomposition is closely related to your problem, see <a href="http://www.cwru.edu/artsci/math/szarek/SzarekICMslides.pdf" rel="nofollow">http://www.cwru.edu/artsci/math/szarek/SzarekICMslides.pdf</a></p>