Laugwitz's conjecture on flat Hessian metrics of Minkowski norms

Let $F$ be a Minkowski norm on $\mathbb{R}^n$, with $n\geq 3$, and let

$$g=\tfrac12{\rm d}^2F^2$$

be its Hessian metric on $\mathbb{R}^n\setminus\{0\}$. Laugwitz's conjecture. If $g$ is flat on $\mathbb{R}^n\setminus\{0\}$, then $F$ is Euclidean. This conjecture extends the corresponding theorem for absolutely homogeneous Minkowski norms; the paper proves it in the setting of $SO(k)\times SO(n-k)$-symmetric Minkowski norms.