Affinely Regular Polygons as Extremals of Area Functionals

For any convex n-gon P we consider the polygons obtained dropping a vertex or an edge of P . The area distance of P to such (n − 1)-gons, divided by the area of P , is an affinely invariant functional on n-gons whose maximizers coincide with the affinely regular polygons. We provide a complete proof of this result. We extend these area functionals to planar convex bodies and we present connections with the affine isoperimetric inequality and parallel X-ray tomography.