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?



Critical Radius for Infinite Dimensional Sphere Packing
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.


