Proof of Laugwitz Conjecture and Landsberg Unicorn Conjecture for Minkowski norms with -symmetry

Abstract

AbstractFor a smooth strongly convex Minkowski norm$F:\mathbb {R}^n \to \mathbb {R}_{\geq 0}$, we study isometries of the Hessian metric corresponding to the function$E=\tfrac 12F^2$. Under the additional assumption thatFis invariant with respect to the standard action of$SO(k)\times SO(n-k)$, we prove a conjecture of Laugwitz stated in 1965. Furthermore, we describe all isometries between such Hessian metrics, and prove Landsberg Unicorn Conjecture for Finsler manifolds of dimension$n\ge 3$such that at every point the corresponding Minkowski norm has a linear$SO(k)\times SO(n-k)$-symmetry.