  Using suitable convex functions, we construct a new family of flat Minkowski
planes whose automorphism groups are at least $3$-dimensional. These planes
admit groups of automorphisms isomorphic to the direct product of $\mathbb{R}$
and the connected component of the affine group on $\mathbb{R}$. We also
determine isomorphism classes, automorphisms and possible Klein-Kroll types for
our examples.
