Area-preserving \(C^ 1\)-mappings are isometries (Q1373289)

A linear map \(L : E^n \to E^n\) of the Euclidean space \(E^n\) which preserves the \(k\)-dimensional volume (for some fixed \(1\leq k\leq n-1\)) is an isometry of \(E^n\). This was proved by \textit{J. Firey} [Am. Math. Mon. 72, 645 (1965; Zbl 0134.16401)]. Ten years later, \textit{P. McMullen} [Elem. Math. 30, 86-87 (1975; Zbl 0307.50004)] showed that for \(k=n-1\) the linear map \(L\) can be replaced by a homeomorphism. It is not known whether or not this generalization is true for other values of \(k\). Using elementary methods only the author provides a short and elegant proof of the following generalization of Firey's theorem: A \(C^1\)-map \(L : E^n \to E^n\) of the Euclidean space \(E^n\) which preserves the \(k\)-dimensional volume of all \(k\)-balls for some fixed integer \(k\) with \(1\leq k\leq n-1\) is an isometry of \(E^n\).