On isometric reflections in Banach spaces

We obtain the following characterization of Hilbert spaces. Let $E$ be a Banach space the unit sphere $S$ of which has a hyperplane of symmetry. Then $E$ is a Hilbert space iff any of the following two conditions is fulfilled: a) the isometry group $\operatorname{Iso}E$ of $E$ has a dense orbit in $S'$ ; b) the identity component $G_0$ of the group $\operatorname{Iso}E$ endowed with the strong operator topology acts topologically irreducible on $E$. Some related results on infinite dimensional Coxeter groups generated by isometric reflections are given which allow us to analyse the structure of isometry groups containing sufficiently many reflections.