Negative expected value for iid standard normal random variables I would like to show analytically that:
$$\Bbb{E}\left[\frac{(X_2-X_1)(Y_2-Y_1)}{|X_2+Y_2-X_1-Y_1|}\right] < 0$$
I would be happy just showing that this is true when $(X_1,Y_1,X_2,Y_2)$ are iid standard normal random variables, but a more general demonstration would be even better.
Montecarlo integration shows the expectation to be strongly negative (much less than zero) for pretty much any distribution governing the $X_1, Y_1, X_2, Y_2$.
 A: Using the hint of Michael Hardy and letting $X = \frac{X_2 - X_1}{\sqrt{2}}$ and $Y = \frac{Y_2 - Y_1}{\sqrt{2}}$, $$\frac{(X_2-X_1)(Y_2-Y_1)}{|X_2+Y_2-X_1-Y_1|} = \frac{\sqrt{2} XY}{|X+Y|}$$
So it suffices to show that $$\mathbb{E} (Z) < 0$$ where $Z =  \frac{XY}{|X+Y|}$ and $X, Y \sim N(0, 1)$.
An intuitive explanation first: For $Z > 0$ to be true, $X$ and $Y$ have to be of the same sign. Because they are of the same sign, this makes $|X+Y|$ larger (on average) and the overall "weight" of $Z$ smaller. On the other hand for $Z < 0$ to be true, $X$ and $Y$ have to have different signs. Because of this, $|X+Y|$ is smaller (on average), and the weight of $Z$ is larger. This makes the overall expected value negative.
A more mathematical explanation: The expected value of $Z$ is $$\int_{-\infty}^\infty \int_{-\infty}^\infty f_X(x) f_Y(y) \frac{xy}{|x+y|} \, dx \, dy$$
Because $X$ and $Y$ are symmetric about $0$, this can be rewritten as $$\int_0^\infty \int_0^\infty f_X(x) f_Y(y) \left(\frac{xy}{x+y} + \frac{(-x)(-y)}{|-x-y|} + \frac{(-x)y}{|(-x)+y|} + \frac{x(-y)}{|x+(-y)|}  \right) dx \, dy$$
The integrand can be simplified to $$2xy f_X(x) f_Y(y)\left(\frac{1}{x+y} - \frac{1}{|x-y|} \right)$$
It is easy to see that $\frac{1}{x+y} - \frac{1}{|x-y|} < 0$ for $x, y > 0$ since $$|x-y| < x+y $$ $$\frac{1}{x+y}<\frac{1}{\left|x-y\right|}$$ $$\frac{1}{x+y}-\frac{1}{\left|x-y\right|}<0$$
and so the integrand will be negative. Therefore the expected value of $Z$ will be negative. A similar explanation would work for any $X, Y$ that are both symmetric about $x = 0$ and thus also any $X_1, X_2, Y_1, Y_2$ such that $X_1$, $X_2$ are iid and $Y_1, Y_2$ are iid ($X_1, X_2, Y_1, Y_2$ don't have to be iid).
