Chris Landauer wrote: [...]
Now the program just uses a simple radius bounded backtracking search for the points in the polygon, with some special processing near the end.
Once the area of the polygon determined by the current set of points exceeds your bound, you can cut off the search. Perhaps it's obvious that you already do this, since you can run the search at all. How about this: Start by picking the vertex V most distant from the origin. All convex polygons with area A and including {O, V} must lie between the two lines parallel to OV at a distance of 2A/|V|. That will probably be only about 4A points. I'm trying to figure out what to do with SL(2,Z). Suppose all points of the polygon lie within a rectangle M1 <= x <= M2 and 0 <= y <= N <= M2-M1. Under what circumstances will the map (x,y) -> x-y,y) or (x,y) -> (x+y,y) reduce the value of M2-M1? If M2-M1 goes below N, we can transpose and try again. Can this be made to map to a polygon with a small radius bound? What gets me is that Jamie Simpson apparently proved that the minimum must be at least 39/2. Is this something that can be pushed with computation? Dan