Extremal antipodal polygons and polytopes

2013/01/28 by O. Aichholzer, L. E. Caraballo, Aichholzer, O. +9
Computer Science · Mathematics · #Combinatorics (math.CO) #Computational Geometry (cs.CG) #FOS: Computer and information sciences #FOS: Mathematics #Metric Geometry (math.MG) #cs.CG #math.CO #math.MG

paper · pdf · doi:10.48550/arxiv.1301.6667

arxiv created 2013/01/28 · arxiv updated 2013/01/29

Abstract

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.

Related

• A cubical antipodal theorem
• Higher rank antipodality
• Asymmetric polygons with maximum area
• Explicit constructions of centrally symmetric k-neighborly polytopes and large strictly antipodal sets
• Weight Balancing on Boundaries