Critical Radius for Infinite Dimensional Sphere Packing - MathOverflow

Critical Radius for Infinite Dimensional Sphere Packing

2011-04-13T07:30:10Z

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 $r<1$ inside.

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.

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?

Answer by Mikael de la Salle for Critical Radius for Infinite Dimensional Sphere Packing

2011-04-13T09:31:54Z

Your value of $r$ is the best.

Equivalently, $\rho=\sqrt 2$, where $\rho$ is the sup, over all infinite sequence $(x_i)$ in the unit ball of a Hilbert space, of $\inf_{i\neq j} |x_i-x_j|$.

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 $\langle x_i,e\rangle<0$, and even that $\langle x_i,e\rangle<-\delta$ 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.

Answer by Pietro Majer for Critical Radius for Infinite Dimensional Sphere Packing

2011-04-13T20:42:02Z

The optimality of your configuration can be shown as a plain consequence of the Kirszbraun theorem.

(I happened to ask myself this problem too, and eventually added this short section 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).