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((.)) phi-convex (i.e., it satisfies a kind of exterior sphere condition with locally uniform radius), provided T((.)) 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((.)) and local estimates on the total variation of DT.