>Assume $\Gamma$ acts by isometries on a separable Hilbert space $H$, and 
$$\operatorname{diam} H/\Gamma\le 1.$$ 
Is it true that $H/\Gamma$ is compact?


**Comment.**

At the moment I do not have an answer even if $\Gamma$ acts by translations. Here is a related question: 
>Let $L$ be a lattice in $\mathbb R^q$ ($q$ is any positive integer). 
Assume $$\operatorname{diam} H/L>1000.$$ 
Is it true that there is a midpoint $m$ of two points in $L$ such that $|m-x|>1$ for any $x\in L$?

If the answer to the this question is "YES" then the answer to my original question is "NO".

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**Stupid example.** Assume the action of $\Gamma$ on $H=\ell_2$ is generated by coordinate translations $x_n\mapsto x_n+\epsilon_n$. Then 
$$\operatorname{diam} H/\Gamma=\tfrac12\cdot\sqrt{\sum_{n=1}^\infty\epsilon_n^2}.$$ 
Thus, if $\operatorname{diam} H/\Gamma\le 1$ then $H/\Gamma$ is a quotient of Hilbert cube, and has to be compact.