geodesics on a Finsler space

Question

A positive $1$-dimensional parametric integrand (short $1$-p.i.) is a continuous function on the tangent space of a manifold $F:TM\rightarrow \mathbb{R}_{\geq 0}$ that is positively homogeneous. This gives a variational problem involving $1$-dimensional manifolds (curves) or more generally $1$-integer multiplicity currents. A $1$-p.i. defined on TU, where U is an open set in R^n is called semi-elliptic if straight lines are locally minimizing. In the book of Krantz and Parks, Geometric integration Theory, it is proven that if $F$ is convex in the tangent variable for every $x\in U$ then it is semi-elliptic. Now take any Finsler norm on $TU=U\times \mathbb{R}^n$. It is strictly convex in the tangent variable (see Th. 1.2.2 in the book Bao+Chern+Shen). By definition the solutions of the associated variational problem are geodesics.

Combining the two results we get that straight lines are always geodesics. Where is my mistake?

Answer 1

Probably you are misinterpreting some statement in Krantz and Parks, since the claim that if $F$ is convex in the tangent variable for every $x\in U$ then straight lines are locally minimizing is clearly false. Either you are misunderstanding their definition of 'semi-elliptic' or else you are missing some other assumption about $F$ in their sufficient condition for 'semi-ellipticity'. (I'm not familiar with their book, so I can't say which it is.) For example, the length integrand for most Riemannian metrics will give you counterexamples to the 'minimizers are straight lines' conclusion.