Mollifier smoothings of strongly convex $C^0$-Finsler structures

Let $M$ be a smooth manifold and $TM$ its tangent bundle. A $C^0$-Finsler structure of $M$ is a continuous function $F:TM\to [0,\mathrm{\infty })$ such that restricted to each tangent space is an asymmetric norm. A mollifier smoothing of $F$ is a family of Finsler structures $F_{\epsilon }:TM\to [0,\mathrm{\infty })$, parameterized by $\epsilon >0$ and constructed using the standard mollifier, such that $F_{\epsilon }\to F$ uniformly on compact subsets as $\epsilon \to 0$. In this work, we construct two mollifier smoothings of $F$. In the first, we assume that $F:TM\to [0,\mathrm{\infty })$ is strongly convex, that is, that $F$ restricted to each tangent space is strongly convex. In the second, we assume that $M=G$ is a Lie group and that $F:TG\to [0,\mathrm{\infty })$ is strongly convex and left-invariant, and in this case, $F_{\epsilon }$ is also left-invariant. In both cases, we prove that if $F$ is a Finsler structure, objects such as the fundamental tensor, the Chern, Cartan, Hashiguchi and Berwald connections, and the flag curvature of $(M,F_{\epsilon })$ converge uniformly on compact subsets to the corresponding objects of $(M,F)$ as $\epsilon \to 0$.