  In this paper we study isometry-invariant Finsler metrics on inner product
spaces over $\mathbb{R}$ or $\mathbb{C}$, i.e. the Finsler metrics which do not
change under the action of all isometries of the inner product space. We give a
new proof of the analytic description of all such metrics. In this article the
most general concept of the Finsler metric is considered without any additional
assumptions that are usually built into its definition. However, we present
refined versions of the described results for more specific classes of metrics,
including the class of Riemannian metrics. Our main result states that for an
isometry-invariant Finsler metric the only possible linear maps under which the
metric is invariant are scalar multiples of isometries. Furthermore, we
characterise the metrics invariant with respect to all linear maps of this
type.
