Doubling dimension of a Euclidean space - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T18:04:42Z http://mathoverflow.net/feeds/question/100777 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/100777/doubling-dimension-of-a-euclidean-space Doubling dimension of a Euclidean space Valerio Capraro 2012-06-27T13:54:43Z 2012-06-28T17:08:30Z <p>The doubling dimension of a metric space $X$ is the smallest positive integer $k$ such that every ball of $X$ can be covered by $2^k$ balls of half the radius.</p> <p>It is well known that the doubling dimension $d(n)$ of the Euclidean space $\mathbb R^n$ is $O(n)$, which means that there is a constant $C$ such that for large $n$ one has $d(n)\leq Cn$. A posteriori, I can find a new constant $D$ that works for all $n$. I would like to have an explicit description of this new constant. In other words,</p> <p><strong>Question:</strong> What explicit and possibly nice and small constant $D>0$ would guarantee that $d(n)\leq Dn$, for all $n$?</p> <p><strong>Edit.</strong> As observed by Igor Rivin, $D=\log 2$ should be good for $n\geq7$, by a theorem of Verger-Gaugry. Any idea for all $n$? I have to clarify that at the moment I am not interested in the best possible constant, but in <em>some good-looking</em> constant, something to make aesthetically pleasant a certain formula that I found out.</p> <p>Thank you in advance,</p> <p>Valerio</p> http://mathoverflow.net/questions/100777/doubling-dimension-of-a-euclidean-space/100780#100780 Answer by Igor Rivin for Doubling dimension of a Euclidean space Igor Rivin 2012-06-27T14:04:25Z 2012-06-27T14:04:25Z <p>As shown in <a href="http://dl.dropbox.com/u/5188175/ballcover.pdf" rel="nofollow">this paper</a>,Theorem 1.2, $D \leq \log 2.$ I remark that this paper came up in my answer to <a href="http://mathoverflow.net/questions/98007/covering-a-unit-ball-with-balls-half-the-radius" rel="nofollow">this question</a>, and there is a bug for small $n$ ($n &lt; 7$), but the author's interest was apparently similar to yours, so the large $n$ results should be correct. (the paper is: "Covering a Ball with Smaller Equal Balls in \$\mathbb{R}^n," by Jean-Louis Verger-Gaugry)</p>