Let S be a set of 2n points on a circle such that for each point p 2 S also its antipodal (mirrored with respect to the circle center) point p 0 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 0 ) of S. We provide a complete characterization of antipodal polygons which maximize (minimize, respectively) the area among all antipodal polygons ofS. 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.
This work contains almost 30,000 articles containing over 25 million words on musicians, composers, musicologists, instruments, places, genres, terms, performance practice, concepts, acoustics and more. All the articles are written by experts in their subject. There are over 500 biographies of composers, performers and writers on music and over 1,500 articles on styles, terms, and genres. It also includes: over 500 articles on ancient music and church music over 700 articles on regions, countries and cities over 2,000 articles on instruments and their makers and performance practice over 650 articles on printing and publishing over 1,200 articles on world music over 1,000 articles on popular music, light music and jazz over 250 articles on concepts 85 articles on acoustics 126 articles on sources and a one volume index.