Deforming convex bodies in Minkowski geometry

TL;DR

This paper introduces a new class of Minkowski norm deformations in Minkowski geometry, generalizing known constructions and analyzing their impact on Cartan torsion, with implications for Finsler geometry.

Contribution

It defines a novel deformation $T_{\bf b,\phi}$ of Minkowski norms using linearly independent 1-forms and a smooth function, extending the class of computable Minkowski norms.

Findings

Deformation $T_{\bf b,\phi}$ produces Minkowski norms with rotation hypersurface indicatrices.

Characterization of when Cartan torsions of original and deformed norms coincide.

Establishment of an equivalence relation on Minkowski norms via compositions of $T_{\bf b,\phi}$.

Abstract

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.