Assume that \( A_1A_2 = D \) is the maximum distance between two points in the set. Let \( d \) be the maximum distance between a point in the set and the line \( A_1A_2 \). Then, there exists a triangle with area at least \( \frac{1}{2} d D \). Notice that all vertices must lie within the region bounded by the circles centered at \( A_1 \) and \( A_2 \) with radius \( D \), and the two parallel lines that are a distance \( d \) away from \( A_1A_2 \). Clearly, the polygon lies within this region, which has an area strictly smaller than \( 2dD \). The result of the problem follows.