Summary

• The lowest upper bound for the area ratio is 1 in convex n-gons with n > 5.

Use a Degenerate Case to Find the Maximum

For convex polygons with six or more sides, the ratio of the area of the midpoint pentagon to the area of the original is as close to 1 as we please. We can achieve this by placing the first two vertices near one vertex of a triangle, the next two near the second, and all the rest near the third. The ratio cannot be larger than 1 since the midpoint polygon lies inside the original convex polygon.