Critical Radius for Infinite Dimensional Sphere Packing - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-26T02:55:55Z http://mathoverflow.net/feeds/question/61513 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/61513/critical-radius-for-infinite-dimensional-sphere-packing Critical Radius for Infinite Dimensional Sphere Packing Ryan Thorngren 2011-04-13T07:30:10Z 2011-04-13T21:19:20Z <p>Hello. I'd like to consider the open unit ball in an infinite dimensional Hilbert space and ask when can we fit infinitely many open balls of radius $r&lt;1$ inside.</p> <p>For example, when $r=1/(1+\sqrt2)$, we can pick an orthonormal basis $(x_1,...)$ for our Hilbert space and put the centers of the balls at $(1-r)x_i = \sqrt2/(1+\sqrt2)x_i$ for each $i$. The distance between any two centers is thus $\sqrt2/(1+\sqrt2)\sqrt2 = 2r$, so indeed the balls just kiss each other.</p> <p>Can we fit any larger balls? What is the critical radius $r_\infty$ such that for $r>r_\infty$ we may only fit finitely many balls of radius $r$, but for smaller $r$ we may fit infinitely many?</p> http://mathoverflow.net/questions/61513/critical-radius-for-infinite-dimensional-sphere-packing/61527#61527 Answer by Mikael de la Salle for Critical Radius for Infinite Dimensional Sphere Packing Mikael de la Salle 2011-04-13T09:31:54Z 2011-04-13T09:31:54Z <p>Your value of $r$ is the best.</p> <p>Equivalently, $\rho=\sqrt 2$, where $\rho$ is the sup, over all infinite sequence $(x_i)$ in the unit ball of a Hilbert space, of $\inf_{i\neq j} |x_i-x_j|$.</p> <p>Here is a proof, by contradiction. Assume that $\rho>\sqrt 2$ and pick a sequence $(x_i)$ such that $\inf_{i\neq j} |x_i-x_j|$ is almost $\rho$. Take $e$ the unit vector $x_1/|x_1|$. Then from the inequality $|x_i-x_1|>\sqrt 2$ we get that $\langle x_i,e\rangle&lt;0$, and even that $\langle x_i,e\rangle&lt;-\delta$ for some positive $\delta$ depending on $\rho$ only. In particular, every element in the sequence $(x_2,x_3,...)$ belongs to the ball of radius $1-\epsilon$ around $-\delta e$ for some $\epsilon>0$ depending on $\rho$ only. This implies that $\inf_{i\neq j >1} |x_i-x_j| \leq (1-\epsilon)\rho$. But $\inf_{i\neq j} |x_i-x_j|$ was arbitraly close to $\rho$. We thus get $\rho \leq (1-\epsilon)\rho$, a contradiction.</p> http://mathoverflow.net/questions/61513/critical-radius-for-infinite-dimensional-sphere-packing/61590#61590 Answer by Pietro Majer for Critical Radius for Infinite Dimensional Sphere Packing Pietro Majer 2011-04-13T20:42:02Z 2011-04-13T21:19:20Z <p>The optimality of your configuration can be shown as a plain consequence of the Kirszbraun theorem.</p> <p>(I happened to ask myself this problem too, and eventually added <a href="http://en.wikipedia.org/wiki/Packing_problem#Spheres_into_a_Euclidean_ball" rel="nofollow">this short section</a> in a wiki article, thinking that it could be useful one day --not completely true, since your question has been already answered by Mikael de la Salle). </p>