A large family of projectively equivalent $C^0$-Finsler manifolds

Let $M$ be a differentiable manifold and $TM$ be its tangent bundle. A $C^0$-Finsler structure on $M$ is a continuous function $F: TM \rightarrow \mathbb R$ such that its restriction to each tangent space is a norm. In this work we present a large family of projectively equivalent $C^0$-Finsler manifolds $(\widehat{M}\cong {\mathbb{R}}^2,\widehat{F})$. Their structures $\widehat{F}$ don’t have partial derivatives and they aren’t invariant by any transformation group of $\widehat{M}$. For every $p,q\in (\widehat{M},\widehat{F})$, we determine the unique minimizing path connecting $p$ and $q$. They are line segments parallel to the vectors $(\sqrt{3}\mathrm{/}2,1\mathrm{/}2)$, $(0,1)$ or $(-\sqrt{3}\mathrm{/}2,1\mathrm{/}2)$, or else a concatenation of two of these line segments. Moreover $(\widehat{M},\widehat{F})$ aren’t Busemann $G$-spaces and they don’t admit any bounded open $\widehat{F}$-strongly convex subsets. Other geodesic properties of $(\widehat{M},\widehat{F})$ are also studied.