last week I found a question during my seminar work and couldn't solve it. I'm wondering if anybody else wants to work on this since the question is pretty easy but a solution turns out to be (at least) not as easy as the question. So,
Given function values y_i for finite x_i's in the interval (0,1) such that the frequency polygon can be convex. What is the convex function with the least area connecting the points (0,0) and the (x_i,y_i)'s?
I had different ideas, the most important (and somewhat only useful) is that I can look in the class of piecewise linear functions for the 'solution'. Furthermore, I should have n pieces and the pieces are somehow centered at (x_i,y_i).