# Probability theoretic inequality

1. Jul 21, 2008

### winterfors

Can anyone help me prove under what conditions on the distance function $$d(x_1,x_2)$$ the following inequality holds for any two probability distributions (represented by probability densities) $$p(x)$$ and $$q(x)$$ :

$$2\int{\int{d^2(x_1,x_2)p(x_1)q(x_2)dx_1dx_2} \geq \int{\int{d^2(x_1,x_2)p(x_1)p(x_2)dx_1dx_2} + \int{\int{d^2(x_1,x_2)q(x_1)q(x_2)dx_1dx_2}$$

where $$d^2(x_1,x_2)$$ is the squared distance between $$x_1$$ and $$x_2$$ in some metric space $$\Theta$$. All integrals are over $$\Theta$$.

One can easily verify by insertion that the inequality holds for a Euclidian metric where $$d^2(x_1,x_2)=(x_1-x_2)^2$$, with equality if and only if the expectation of $$p(x)$$ and $$q(x)$$ are the same.

It must surely hold for some more general class of metrics (described by $$d^2(x_1,x_2)$$) - possibly all metrics - but I've so far failed to demonstrate it. Does anyone have an idea of how to prove it in some more general case?