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arXiv · 1301.6667

Extremal antipodal polygons and polytopes

O. Aichholzer·L. E. Caraballo·J. M. Díaz-Báñez·R. Fabila-Monroy·C. Ochoa·P. Nigsch
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Abstract

Let $S$ be a set of $2n$ points on a circle such that for each point $p \in S$ also its antipodal (mirrored with respect to the circle center) point $p'$ belongs to $S$. A polygon $P$ of size $n$ is called \emph{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.

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O. Aichholzer, L. E. Caraballo, J. M. Díaz-Báñez, R. Fabila-Monroy, C. Ochoa, P. Nigsch. 2013-01-28. Extremal antipodal polygons and polytopes. https://arxiv.org/abs/1301.6667

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