Some New Regularity Properties for the Minimal Time Function

A minimal time problem with linear dynamics and convex target is considered. It is shown, essentially, that the epigraph of the minimal time function $T(\cdot)$ is $\varphi$-convex (i.e., it satisfies a kind of exterior sphere condition with locally uniform radius), provided $T(\cdot)$ is continuous. Several regularity properties are derived from results in [G. Colombo and A. Marigonda, Calc. Var. Partial Differential Equations, 25 (2005), pp. 1-31], including twice a.e. differentiability of $T(\cdot)$ and local estimates on the total variation of $DT$.