I'm working on a problem in discrete geometry, more specifically on visibility of polygons. The easiest instance of this problem reduces to the following.
Among all density functions $f:[0,1]\to R$, find the one in which the following is minimized: $$ \int_{0}^{1} x^2 f(x) \biggl( \int_x^1 f(y)dy \biggr) dx + \int_0^1 f(x) \biggl(\int_{x}^1 f(y)(1-y^2) dy\biggr) dx $$
I'm pretty sure that the optimum is attained when $f$ is the constant function $f(x)=1$. (This is also backed up by geometric considerations). However, my knowledge of calculus of variations is very modest. I have spent a couple of days searching for ideas, theorems that can be applied to this problem, with absolutely no success so far.
I would appreciate any ideas on how to approach this problem. If this turns out to be out of my reach (because it requires rather sophisticated techniques that could take me a lot of time to master) I would be glad to offer co-authorship to someone for which this question is not out of reach. As I said at the beginning, this is the easiest instance of the problem. The general problem is of the same flavour, and for that one I happen to "know'' (again, from geometrical considerations, and plenty of experiments) that the optimum is attained with density functions that involve Dirac delta functions.
Any help will be greatly appreciated.
