On the perimeters of simple polygons contained in a plane convex body

TL;DR

This paper determines the maximum perimeter of simple n-gons within any convex shape in Euclidean or hyperbolic planes, extending previous results from disks to general convex bodies.

Contribution

It generalizes the known perimeter bounds from disks to arbitrary convex bodies in both Euclidean and hyperbolic geometries.

Findings

Maximum perimeter achieved by specific n-gons in convex bodies.

Extension of previous disk-based results to general convex shapes.

Applicable to both Euclidean and hyperbolic planes.

Abstract

A simple n-gon is a polygon with n edges such that each vertex belongs to exactly two edges and every other point belongs to at most one edge. Brass, Moser and Pach asked the following question: For n > 3 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 supremum is the perimeter of an isosceles triangle inscribed in the disk, with an edge of multiplicity n-2. L\'angi generalized their result for polygons contained in a hyperbolic disk. In this note we find the supremum of the perimeters of simple n-gons contained in an arbitrary plane convex body in the Euclidean or in the hyperbolic plane.