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

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