TITLE: Is a space with p-norm a Finsler manifold?
QUESTION [5 upvotes]: Suppose $\mathbb{R}^n$ is equipped with the p-norm $\left\Vert x \right\Vert_p$. Let $x\in \mathbb{R}^n$ and let $y$ be in a neighborhood of $x$. The distance between $x$ and $y$ can be defined as $\left\Vert x-y \right\Vert_p$. Is this setting an example of a Finsler manifold? If so, it should be a very simple example of a Finsler manifold. Can you point out some references about the study of the geometry of this type of p-norm spaces, like the curvature and geodesics? (Apparently when $p=2$, it is the trivial case of a Euclidean space, which is a trivial case of Riemannian manifold, which is a special case of Finsler manifold. But what about other $p\neq 2$?)

REPLY [4 votes]: It is a Finsler metric for $p>1$. I don't know a reference. But you can easily see that its indicatrix is smooth and strictly convex and symmetric. The triangle inequality says that the geodesics are the usual straight lines. The Finsler flag curvature vanishes by the Jacobi equation. I don't know the complete description of the curvature.