Deforming convex bodies in Minkowski geometry

Abstract

We introduce and study deformation $T_{\mathbf{b},\varphi }$ of Minkowski norms in ${\mathbb{R}}^n$, determined by a set $\mathbf{b}=({\beta }_1,\dots ,{\beta }_p)$ of linearly independent 1-forms and a smooth positive function $\varphi$ of $p$ variables. In particular, the $T_{\mathbf{b},\varphi }$-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 $T_{\mathbf{b},\varphi }$-deformations with $\mathbf{b}$'s of length $p$ to define an equivalence relation $\stackrel{p}{\sim }$ 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 $T_{\mathbf{b},\varphi }$-image either coincide or differ by a $C$-reducible term.

12 pages, 3 figures