We introduce and study deformation $T_{ { \bf b},\phi}$ of Minkowski norms in $\mathbb{ R}^n$, determined by a set ${ \bf b}=(\beta_1,\ldots,\beta_p)$ of linearly independent 1-forms and a smooth positive function $\phi$ of $p$ variables. In particular, the $T_{ { \bf b},\phi}$-image of a Euclidean norm $\alpha$ is a Minkowski norm, whose indicatrix is a rotation hypersurface with a $p$-dimensional axis passing through the origin. For $p=1$, our deformation generalizes construction of $(\alpha,\beta)$-norm; the last ones form a rich class of "computable" Minkowski norms and play an important role in Finsler geometry. We use compositions of $T_{ { \bf b},\phi}$-deformations with ${ \bf b}$'s of length $p$ to define an equivalence relation $\overset{ p}\sim$ on the set of all Minkowski norms in $\mathbb{ R}^n$. We apply M. Matsumoto result to characterize the cases when the Cartan torsions of a norm and its $T_{ { \bf b},\phi}$-image either coincide or differ by a $C$-reducible term.
Categories: math.DG