Area-perimeter duality in polygon spaces

Exact solution of the staircase and row-convex polygon perimeter and area generating function

An explicit expression is obtained for the perimeter and area generating function G(y, z) = ∑ n>=2 ∑ m>=1 cn,my z, where cn,m is the number of row-convex polygons with area m and perimeter n. A similar expression is obtained for the area-perimeter generating function for staircase polygons. Both expressions contain q-series.

Polygon Spaces and Grassmannians

We study the moduli spaces of polygons in R and R, identifying them with subquotients of 2-Grassmannians using a symplectic version of the Gelfand-MacPherson correspondence. We show that the bending flows defined by Kapovich-Millson arise as a reduction of the Gelfand-Cetlin system on the Grassmannian, and with these determine the pentagon and hexagon spaces up to equivariant symplectomorphism....

Ranking efficient units by regular polygon area in DEA

Finding the Maximum Area Centrosymmetric Polygon in a Convex Polygon

Homology of Planar Polygon Spaces

In this paper we study topology of the variety of closed planar n-gons with given side lengths l1, . . . , ln. The moduli space M` where ` = (l1, . . . , ln), encodes the shapes of all such n-gons. We describe the Betti numbers of the moduli spaces M` as functions of the length vector ` = (l1, . . . , ln). We also find sharp upper bounds on the sum of Betti numbers of M` depending only on the n...