Chris Landauer wrote:

Similary, I noticed that among all of the minimum area polygons that my program found for even n, all of them were centrally symmetric (that is, the second half edge directions are the negatives of the first half ones). I think that there is a simple proof of this (perhaps less cryptic and subscripty than the one in Barany and Tokushiga)....

Define an "extremal pair of vertices" of a convex polygon to be a pair of vertices that are the intersection of the polygon with two parallel lines. For each vertex v there is at least one vertex v' such that v, v' is an extremal pair; in a centrally symmetric polygon there is exactly one. Consider the extremal pairs (v,v'), taking vertices v in order around the polygon, and taking v' in the same order for each v. Let f(v,v') be the the number of vertices from v to v' (counting in the same order, and counting only one of v,v'). f(v,v') can increase or decrease by at most one from one pair to the next, and f(v,v')+f(v',v)=2n. So there must be some pair for which f(v,v')=n. Take such a pair, cut the polygon in two parts through the line vv', and paste two copies of one part together. This is a centrally convex polygon, and if we never choose the larger of two unequal parts the new polygon will have area at most that of the original polygon. So for any minimal area 2n-gon, we can find a symmetric 2n-gon with the same area. I'm surprised, though, to hear that _all_ minimal 2n-gons were symmetric. I would have expected some chimerae--mismatched pairs of equal area. Perhaps there's something else going on here. I will say that I have not been able to get through even the third page of Barany&Tokushiga. I think they are talking about the relationship between a centrally-symmetric polygon and the convex hull of its edge vectors, but I haven't been able to read it in a way that looks believable. They write "Area P(C)" but I'm not sure what P(C) is supposed to mean; also they write "A(C)" but they only define A(n)--is that supposed to mean Area(C)? Anyone care to explain it? The paper is at http://www.renyi.hu/~barany/cikkek.ps/hide.ps . Dan