Suppose $a_1,\dots,a_n$, $b_1,\dots,b_n$, and $c_1,\dots,c_n$ represent three discrete probability measures, so that $a_i,b_i,c_i\geq 0$ for each $i$, and $\sum_{i=1}^n a_i = \sum_{i=1}^n b_i = \sum_{i=1}^n c_i = 1$. I am interested in lower bounding the following sum: $$\sum_{i=1}^n (a_i-b_i)(c_i - b_i). $$
In particular, I am interested in showing that the sum is greater or equal to 0. I believe this to be true, as it has been the case in all the simulations I have run, but I would be equally interested in a counterexample.
Nope: $a_1 = 0, a_2 = 1$, $b_1 = 1/2, b_2 = 1/2$, $c_1 = 1, c_2 = 0$.