Deforming Minkowski norms to Euclidean norms

We study deformations of Minkowski norms with piecewise smooth indicatrices determined by linearly independent $1$-forms and a piecewise smooth positive function. Such a deformation of the Euclidean norm generalizes the classical $(\alpha,\beta)$-norms by M. Matsumoto. We show that any Minkowski norm can be deformed into a Euclidean norm by a composition of such deformations.