Are there any well-known continuous area-preserving maps between fat convex polygons and fat rectangles? Specifically, given a fat convex polygon $C$, is there a natural way to choose a fat rectangle $R$ and a continuous mapping $f:C\rightarrow R$ such that area of sub-regions is preserved? I've avoided giving a precise definition of "fatness", but two common definitions are:

*The aspect ratio of the minimum bounding box of $C$ is bounded by some constant (see page 5 of http://portal.acm.org/citation.cfm?id=1137901).

*The ratio between the diameters of the smallest circle containing $C$ and the largest circle contained in $C$ is bounded by some constant (see http://books.google.com/books?id=QS6vnl8WlnQC&pg=PA588).