Let $f:\mathbb{R}\to\mathbb{R}$ a convex decreasing function. Let $x_0 < x_1 < x_2$. Studying the behaviour of the difference quotient, it is clear that $f(x_0)-f(x_2) \leq M (f(x_0)-f(x_1))$ with $M=\frac{x_2-x_0}{x_1-x_0}>0$ .
Now take $F:\mathbb{R}^2\to\mathbb{R}$ convex and decreasing with respect to each variable. Let $x_0 < x_1 < x_2$ and $y_0 < y_1 < y_2$. I ask if a similar condition holds, say for example $F(x_0,y_0)-F(x_2,y_2) \leq M (F(x_0,y_0)-F(x_1,y_1))$ with $M=\max ( \frac{x_2-x_0}{x_1-x_0}, \frac{y_2-y_0}{y_1-y_0} )$ .