On the perimeters of simple polygons contained in a disk

Details On the perimeters of simple polygons contained in a disk On the perimeters of simple polygons contained in a disk

2010-02-20

arxiv.org/abs/1002.3922v1

Mathematics

Metric Geometry

6 pages, 2 figures

Scientific paper

10.1007/s00605-010-0214-z

A simple $n$-gon is a polygon with $n$ edges with each vertex belonging to exactly two edges and every other point belonging to at most one edge. Brass asked the following question: For $n \geq 5$ odd, what is the maximum perimeter of a simple $n$-gon contained in a Euclidean unit disk? In 2009, Audet, Hansen and Messine answered this question, and showed that the optimal configuration is an isosceles triangle with a multiple edge, inscribed in the disk. In this note we give a shorter and simpler proof of their result, which we generalize also for hyperbolic disks, and for spherical disks of sufficiently small radii.

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