Sporadic Reinhardt Polygons
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Sporadic Reinhardt Polygons Kevin G. Hare · Michael J. Mossinghoff
Received: 18 March 2012 / Revised: 26 September 2012 / Accepted: 2 November 2012 / Published online: 6 February 2013 © Springer Science+Business Media New York 2013
Abstract Let n be a positive integer, not a power of two. A Reinhardt polygon is a convex n-gon that is optimal in three different geometric optimization problems: it has maximal perimeter relative to its diameter, maximal width relative to its diameter, and maximal width relative to its perimeter. For almost all n, there are many Reinhardt polygons with n sides, and many of them exhibit a particular periodic structure. While these periodic polygons are well understood, for certain values of n, additional Reinhardt polygons exist, which do not possess this structured form. We call these polygons sporadic. We completely characterize the integers n for which sporadic Reinhardt polygons exist, showing that these polygons occur precisely when n = pqr with p and q distinct odd primes and r ≥ 2. We also prove that a positive proportion of the Reinhardt polygons with n sides is sporadic for almost all integers n, and we investigate the precise number of sporadic Reinhardt polygons that are produced for several values of n by a construction that we introduce. Keywords Reinhardt polygon · Reinhardt polynomial · Isodiametric problem · Isoperimetric problem · Diameter · Perimeter · Width Mathematics Subject Classification (2000) 11R09 · 52A10 · 52B05
Primary: 52B60 · Secondary:
K. G. Hare Department of Pure Mathematics, University of Waterloo, Waterloo, ON N2L 3G1, Canada e-mail: [email protected] M. J. Mossinghoff (B) Department of Mathematics, Davidson College, Davidson, NC 28036, USA e-mail: [email protected]
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Discrete Comput Geom (2013) 49:540–557
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1 Introduction For a convex polygon in the plane, its diameter is the maximum distance between two of its vertices; its width is the minimal distance between a pair of parallel lines that enclose it. A number of natural problems for polygons arise by fixing the number of sides n, and fixing one of the four quantities diameter, width, perimeter, or area, and then maximizing or minimizing another one of these attributes. Six different nontrivial optimization problems for polygons arise in this way, including, for example, the wellknown isoperimetric problem, where the perimeter of a convex n-gon is fixed, and one wishes to maximize the area. In that case, the regular n-gon is the unique optimal solution for all n, but this is not the case in the other five nontrivial extremal problems in this family. Prior research has shown that a particular family of polygons is optimal in three of these geometric optimization problems, provided that n is not a power of 2: 1. The isodiametric problem for the perimeter (maximize the perimeter, for a fixed diameter). 2. The isodiametric problem for the width (maximize the width, for a fixed diameter). 3. The isoperimetric problem for the width (maximize the width, for a fixed perimeter). Problem
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