  Sequences of polygons generated by performing iterative processes on an
initial polygon have been studied extensively. One of the most popular
sequences is the one sometimes referred to as {\it Kasner polygons}. Given a
polygon $K$, the first Kasner descendant $K'$ of $K$ is obtained by placing the
vertices of $K'$ at the midpoints of the edges of $K$.
  More generally, for any fixed $m$ in $(0,1)$ one may define a sequence of
polygons $\{K^{t}\}_{t\ge 0}$ where each polygon $K^{t}$ is obtained by
dividing every edge of $K^{t-1}$ into the ratio $m:(1-m)$ in the
counterclockwise (or clockwise) direction and taking these division points to
be the vertices of $K^{t}$.
  We are interested in the following problem
  {\it Let $m$ be a fixed number in $(0,1)$ and let $n\ge 3$ be a fixed
integer. Further, let $K$ be a convex $n$-gon and denote by $K'$, the first
$m$-Kasner descendant of $K$, that is, the vertices of $K'$ divide the edges of
$K$ into the ratio $m:(1-m)$. What can be said about the ratio between the area
of $K'$ and the area of $K$, when $K$ varies in the class of convex $n$-gons?}
  We provide a complete answer to this question.
