On the areas of midpoint polygons

For a polygon V₁...V_n in the Euclidean plane, let V₁ ¹...V_n ¹ denote its midpoint polygon. By induction, its m-th midpoint polygon V₁ ^m...V_n ^m is defined to be the midpoint polygon of V₁ ^m−¹...V_n ^m−¹ . In this paper, we will give different kinds of formulas of the area of V₁ ^m...V_n ^m . We will describe the limit behavior of the area as m goes to infinity, and we will determine the infimum and the supremum of the area among all convex V₁...V_n with a fixed area. Some affine invariants derived from the area will also be discussed.