The VC dimension of convex $d$-gons is $2d+1$. To show that, I can prove the lower bound is $2d+1$. however, I don't know how to prove the upper bound in a rigorous way.
- For low bound, I construct a set of $2d+1$ points, they are all lie on a circle. For a labeling, there are two cases, 1) number of positive labels larger than number of negative labels; 2) number of negative labels larger than number of positive labels. For case 1, using the positive points as the vertices of polygon. For case 2, using the tangent lines of circle on positive points as the edges of polygon. Then this set of points is shattered by convex $d$-gon.
- For upper bound, My intuitive idea is, for any set of $2d+2$ points and they are labeled as $d+1$ positive, $d+1$ negative. If all the positive points lie on a polygon, then there is at least a pair of points are co-linear, but this is not the case in our assumption, that is "for any set of $2d+2$ points".
I thought the proof of upper bound is not rigorous enough, so I need some advice and help to make it complete.