TITLE: 2-increasing functions
QUESTION [7 upvotes]: I'm trying to learn about copulas. Two definitions I've come across are the H-volume of a rectangle, which is defined as $V_H=H(x_2,y_2)-H(x_2,y_1)-H(x_1,y_2)+H(x_1,y_1)$. 
The function H, whose domain is a subset of $\mathbb{R}^2$ and whose range is a subset of $\mathbb{R}$, is said to be 2-increasing if $V_H\geq 0$ for all rectangles in the domain of H.
Even though I understand this concept mathematically, can someone give me a graphical/visual interpretation? Is there an easy way to explain, in words, what a 2-increasing function is?

REPLY [2 votes]: It might help to re-write your equation like this:
$V_H=[H(x_1,y_1)+H(x_2,y_2)]-[H(x_1,y_2)+H(x_2,y_1)] \geq 0$, or even better,
$\frac{1}{2}\big[H(x_1,y_1)+H(x_2,y_2)\big] \geq \frac{1}{2}\big[H(x_1,y_2)+H(x_2,y_1)\big]$.  This says that the line from $H(x_1,y_1)$ (the lower left corner) to $H(x_2,y_2)$ (the upper right corner) is at least as high as the line connecting the other two corners, at the point in the plane where their projections cross.