Optimization of Polygons

Since 2010 In recent years, Charles Audet, Pierre Hansen and Fred Messine worked on two families of extremal problems on convex polygons. For a fixed number of sides, each family is to maximize or minimize an attribute such as area, perimeter, diameter, the sum of the distances between all pairs of vertices or the width, by fixing another attribute. In the second family of problems, the polygons are necessarily equilateral, while in the first family, the length of the sides is free.

In continuation of their work, I became interested in solving problems from both families, including the problems of maximizing the perimeter, diameter, area or the sum of the distances of a convex polygon equilateral width unitary.