Let $d:\mathbb R^n\times \mathbb R^n\to\mathbb R^n$ be a distance which induces the Euclidean topology and with which $\mathbb R^n$ is a length space. A continuous curve $\gamma:[a,b]\to\mathbb R^n$ is called a geodesic if it locally minimizes the length. Furthermore, we say that $(\mathbb R^n,d)$ is a projective space if straight lines are geodesics.
I'm wondering if, in a projective space $(\mathbb R^n,d)$, straight lines also globally minimize length, i.e. for every points $x,y\in\mathbb R^n$ we have that $d(x,y) = l(\overline{xy})$, where $l(\overline{xy})$ is the length of the line segment $\overline{xy}$ which connects $x$ and $y$.
I know that in general it is false that local geodesics are also global ones without the hypothesis of projective space, but I can't find a counterexample in this particular case.
My question originates studying a simple version of Hilbert's fourth problem and in particular reading the article Hilbert's fourth problem in two dimension. It seems to me that for the variational approach it is assumed the local definition of the geodesics, whereas for the geometric point of view (see for example Busemann, Alexander and Ambartzumian) segments are assumed to be globally minimizing. However I can't find a relation between the two definitions.
