Area-preserving map between rectangles and fat polygons - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-18T21:09:48Z http://mathoverflow.net/feeds/question/66684 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/66684/area-preserving-map-between-rectangles-and-fat-polygons Area-preserving map between rectangles and fat polygons John Gunnar Carlsson 2011-06-01T19:53:13Z 2011-06-02T07:26:57Z <p>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:</p> <p>*The aspect ratio of the minimum bounding box of $C$ is bounded by some constant (see page 5 of <a href="http://portal.acm.org/citation.cfm?id=1137901" rel="nofollow">http://portal.acm.org/citation.cfm?id=1137901</a>).</p> <p>*The ratio between the diameters of the smallest circle containing $C$ and the largest circle contained in $C$ is bounded by some constant (see <a href="http://books.google.com/books?id=QS6vnl8WlnQC&amp;pg=PA588" rel="nofollow">http://books.google.com/books?id=QS6vnl8WlnQC&amp;pg=PA588</a>).</p> http://mathoverflow.net/questions/66684/area-preserving-map-between-rectangles-and-fat-polygons/66688#66688 Answer by André Henriques for Area-preserving map between rectangles and fat polygons André Henriques 2011-06-01T20:32:33Z 2011-06-01T20:32:33Z <p>This problem is treated in section 18 ("Monge problem for polytopes") of Igor Pak's book: <a href="http://www.math.ucla.edu/~pak/geompol8.pdf" rel="nofollow">http://www.math.ucla.edu/~pak/geompol8.pdf</a> (without any reference to fatness).</p> http://mathoverflow.net/questions/66684/area-preserving-map-between-rectangles-and-fat-polygons/66689#66689 Answer by Dylan Thurston for Area-preserving map between rectangles and fat polygons Dylan Thurston 2011-06-01T20:34:54Z 2011-06-01T20:34:54Z <p>There are lots of ways to do this; without more constraints (or indication of what is desired), it's hard to pick a best one. But here's one. We'll exploit the fact that for any two triangles, there is a unique affine linear map that takes one to the other (with vertices in a specified order); if the two triangles have the same area, this will be area-preserving.</p> <p>Start by chopping up $C$ into triangles $T_1,\dots,T_k$ by drawing lines connecting the vertices to the center.<br> Now take $R$ to be a square with the same area as $C$. Divide up the boundary of $R$ into $k$ intervals $I_1,\dots,I_k$ whose lengths are proportional to the area of the $T_i$. (Some of the intervals may make a turn at the corner.) Divide the square into regions $S_i$ by connecting the ends of $I_i$ to the center of the square. Most of the $S_i$ will be triangles, except for the four at the corners, which are quadrilaterals.</p> <p>Now, the area of $S_i$ is equal to the area of $T_i$, since the area of $S_i$ is proportional to the length of its base (as all $S_i$ have the same height), which is proportional to the area of $T_i$. Now just map $T_i$ to $S_i$ in an area preserving way. For the $S_i$ that are not at the corners, just take the affine linear map from above. For the $S_i$ that do go around the corner, cut both $S_i$ and $T_i$ into two triangles in the same proportion of area, and map the corresponding triangles to each other.</p> http://mathoverflow.net/questions/66684/area-preserving-map-between-rectangles-and-fat-polygons/66714#66714 Answer by Anton Petrunin for Area-preserving map between rectangles and fat polygons Anton Petrunin 2011-06-02T07:26:57Z 2011-06-02T07:26:57Z <p>It is easy to construct a volume preserving map between any two shapes of equal volume <strong>with triangular Jacobian matrix</strong>. (Gromov used such map from the given domain to ball to prove isoperemetric inequality.) </p>