Inequality for convex polygons
Let P be a convex n-gon on the plane. For define as the length of k-th side of P and as the length of projection of P onto the line containing k-th side of the polygon P. Prove that
Firstly, let us prove the first inequality. Indeed, if p is the perimeter of the polygon P, then it's clear that for all . Hence, we obtain
Now, for the second part note that equality holds if, for example, P is a rectangle, so the second inequality is sharp. For polygon P denote
Then, it can be shown that if P′ is polygon which is centrally symmetric to P, then the Minkowski sum satisfy the following equality
Thus, it's sufficient to prove the inequality for Q, i. e. for centrally symmetric polygons (it's well-known that is a centrally symmetric polygon). However, it's quite unclear how to continue this approach.
So, is there any way to end this solution?