  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.
