  Let $K$ be a convex pentagon in the plane and let $K_1$ be the pentagon
bounded by the diagonals of $K$. It has been conjectured that the maximum of
the ratio between the areas of $K_1$ and $K$ is reached when $K$ is an affine
regular pentagon. In this paper we prove this conjecture. We also show that for
polygons with at least six vertices the trivial answers are the best possible.
