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

Lángi, Zsolt (2013) On the perimeters of simple polygons contained in a plane convex body. BEITRÄGE ZUR ALGEBRA UND GEOMETRIE, 54 (2). pp. 643-649. ISSN 0138-4821

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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 et al. (Research Problems in Discrete Geometry, 2005, 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? Audet et al. (Discrete Comput Geom 41:208–215, 2009) 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. Lángi (Monatsh Math 162:61–67, 2011) 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.

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