Enumerating isodiametric and isoperimetric polygons

Abstract

For a positive integer n that is not a power of 2, precisely the same family of convex polygons with n sides is optimal in three different geometric problems. These polygons have maximal perimeter relative to their diameter, maximal width relative to their diameter, and maximal width relative to their perimeter. We study the number of different convex n-gons E ( n ) that are extremal in these three isodiametric and isoperimetric problems. We first characterize the extremal set in terms of polynomials with { − 1 , 0 , 1 } coefficients by investigating certain Reuleaux polygons. We then analyze the number of dihedral compositions of an integer to derive a lower bound on E ( n ) by obtaining a precise count of the qualifying polygons that exhibit a certain periodic structure. In particular, we show that E ( n ) > p 4 n ⋅ 2 n / p if p is the smallest odd prime divisor of n. Further, we obtain an exact formula for E ( n ) in some special cases, and show that E ( n ) = 1 if and only if n = p or n = 2 p for some odd prime p. We also compute the precise value of E ( n ) for several integers by enumerating the sporadic polygons that occur in the extremal set.