Extremal antipodal polygons and polytopes

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.

[1]  Pierre Hansen,et al.  Extremal problems for convex polygons , 2005, J. Glob. Optim..

[2]  Charlotte Truchet,et al.  Computation of words satisfying the "rhythmic oddity property" (after Simha Arom's works) , 2003, Inf. Process. Lett..

[3]  Godfried T. Toussaint,et al.  Computational geometric aspects of rhythm, melody, and voice-leading , 2010, Comput. Geom..

[4]  Alexander Pilz,et al.  New Results on Stabbing Segments with a Polygon , 2013, CIAC.

[5]  Godfried T. Toussaint,et al.  Mathematical Features for Recognizing Preference in Sub-saharan African Traditional Rhythm Timelines , 2005, ICAPR.

[6]  L. Fejes Tóth,et al.  On the sum of distances determined by a pointset , 1956 .

[7]  David P. Dobkin,et al.  Finding Extremal Polygons , 1985, SIAM J. Comput..

[8]  Graham Vulliamy,et al.  The New Grove Dictionary of Music and Musicians , 1982, Popular Music.

[9]  Godfried T. Toussaint,et al.  El Compás Flamenco: A Phylogenetic Analysis , 2004 .

[10]  Esther M. Arkin,et al.  Convex Transversals , 2011, WADS.

[11]  D. Gale.  15. Neighboring Vertices on a Convex Polyhedron , 1957 .

[12]  David Rappaport.  Maximal Area Sets and Harmony , 2007, Graphs Comb..

[13]  Leonidas J. Guibas,et al.  Finding extremal polygons , 1982, STOC '82.

[14]  Raymond Boyd,et al.  African Polyphony and Polyrhythm: Musical Structure and Methodology , 1991 .

[15]  柴田 南雄,et al.  ニューグローヴ世界音楽大事典 = The new Grove dictionary of music and musicians , 1879 .

[16]  J. Matousek,et al.  Using The Borsuk-Ulam Theorem , 2007 .

[17]  D. Benson.  Music: A Mathematical Offering , 2006 .

[18]  Mark Lindley,et al.  The New Grove Dictionary of Music and Musicians , 2001 .