Define the function $F$ on the elements $(x,v)$ of the tangent bundle of $\mathbb{R}^D$ by $$ F(x,v) \triangleq \|x-v\|_2 + \|v\|_1, $$ where $\|\cdot\|_p$ is the $p$-norm on $\mathbb{R}^D$. Does $F$ define a Finsler function (in the sense that the Hessian of $F^2$ is a symmetric bilinear form)?
I can prove that this defines a geodesic length space by convexity arguments but I'm not sure how to show (or check) if it is Finslerian.
