Let $A_1,A_2,\cdots,A_m$ be $m$ subsets of the set of size $n$ . Prove that $$\sum_{i=1}^{m} \sum_{j=1}^{m}|A_i|\cdot |A_i \cap A_j|\geq \frac{1}{mn}\left(\sum_{i=1}^{m}|A_i|\right)^3.$$
Someone know what this inequality background? The article studies such problems? Thank you.
Idea: it seem this inequality with The sum can be written in terms of indicator functions as solve it? How to it?