Characterization of Extremal Antipodal Polygons

Let S be a set of 2n points on a circle such that for each point p ∈ S also its antipodal (mirrored with respect to the circle center) point p′ belongs to S. A polygon P of size n is called antipodal if it consists of precisely one point of each antipodal pair (p, p′) of S. We provide a complete characterization of antipodal polygons which maximize (minimize, respectively) the area among all antipodal polygons of S. Based on this characterization, a simple linear time algorithm is presented for computing extremal antipodal polygons. Moreover, for the generalization of antipodal polygons to higher dimensions we show that a similar characterization does not exist.