Complete characterization of anisotropic geodesics in the Euclidean space

Let $F$ be a lower semicontinuous, 1-homogeneous positive function defined on $\mathbf{R}^n$. We provide a characterization of absolutely continuous paths that minimize the anisotropic $F$-length between two points. The characterization is achieved by establishing a connection between the minimizing paths and the geometry of the anisotropic $F$-isoperimetric set.