Area-perimeter duality in polygon spaces

Abstract

Two natural foliations, guided by area and perimeter, of the configurations spaces of planar polygons are considered and the topology of their leaves is investigated in some detail. In particular, the homology groups and the homotopy type of leaves are determined. The homology groups of the spaces of polygons with fixed area and perimeter are also determined. Besides, we extend the classical isoperimetric duality to all critical points. In conclusion a few general remarks on dual extremal problems in polygon spaces and beyond are given.

Figures (3)

• Figure 1: Regular stars for $n=7$ with their winding numbers.
• Figure 2: Stratification of $\mathbb{R}_{>0}\times \mathbb{R}$.
• Figure 3: Rotating the half line to the horizontal position

Theorems & Definitions (9)

• Theorem 1
• Proposition 1
• Theorem 2
• Lemma 1
• Theorem 3
• Proposition 2
• Theorem 4
• Theorem 5
• Proposition 3