Hello,

Let $V$ an euclidean space, $X$ a finite set of non-zero vertors in V and $\mathcal{H}$ be the set of hyperplanes of the form $a^{\perp}$ for some $a\in X$. Let $W$ be the group generated by reflecions $s_{H}$ for $H\in \mathcal{H}$.

If the set $X$ is invariant under the action of $W$ then is finite. My question is : Does the reciprocal is true ???