On the Perimeters of Simple Polygons Contained in a Plane Convex Body

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 [2, Problem 3, p. 437] asked the following question: For n ≥ 5 odd, what is the maximum perimeter of a simple n-gon contained in a Euclidean unit disk? In 2009, Audet, Hansen and Messine [1] 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. In [3], Lángi 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 convex body in the Euclidean or in the hyperbolic plane.