Chris Landauer wrote:

well, the systematic search program i wrote to find minimum area convex lattice polygons took 20 hours for n=20 and 80 for n=21, with results as follows....

My understanding is that this still does not establish a lower bound on area even for the 11-gon, unless we can somehow prove that there is a minimum-area 11-gon with no edge more than 10 units long. Given the use of SL(2,Z) to force a unit edge, we would have to prove the existence of an 11-gon with a unit edge _and_ no edge longer than 10. I don't know any way of proving such a thing. I gave an argument earlier that would allow us to establish a lower bound rigorously, though I don't know if it's feasible, and I'd like to improve it. I should note that that argument does not allow us to assume a unit edge (since the transform yielding the unit edge might not yield the minimum diameter). Also, in case it wasn't obvious, I was speaking of Euclidean diameter, rather than the Manhattan diameter that Carl uses in his statistics. I've been looking for the net-accessible materials for help with the problem. Steven Finch has a good survey paper http://people.bu.edu/srfinch/cvxl.pdf though I would quibble with his statement that Barany & Tokushige ... computed that a_n lim --- = 0.0185067... n->oo n^3 via a heuristic solution of ~10^10 constrained maximization problems. The paper, which Finch links to at http://www.renyi.hu/~barany/cikkek.ps/hide.ps is more clear in saying that they prove that the actual value of the limit is the minimum of about 10^10 extremal problems (which are too many to solve), and that their heuristic computations indicate that the minimum is "most likely" 0.0185067.... It's possible that Finch means "heuristic" to imply "most likely", but I would be a lot more comfortable if "computed" were changed to "conjectured". Most surprising, to me at least, is that the minimal area n-gon (when mapped via the appropriate linear transformation) seems to approximate an ellipse (x/c)^2 + (y/d)^2 = 1, with c ~ .003573 n^2 and d ~ 1.656 n. Imagine the minimum eccentricity near n=463 . Another item that may be of interest is the fact that we may take the minimum-area 2n-gon to be centrally symmetric, based on a circular descent argument. Dan