I would like to know if the following property (known as the Monotropic Property) still hold or can be extended to the general case of a convex function $f$ : $\mathbb{R}^n$ $\to$ $\mathbb{R}$ ?
$\textbf{Monotropic Property}$
Let $f$ : $\mathbb{R}$ $\to$ $\mathbb{R}$ be a convex function.
If $x_1$, $x_2$, $x_3$ are three scalars such that $x_1$ $\lt$ $x_2$ $\lt$ $x_3$, then $\frac{ f(x_2)−f(x_1)}{x_2−x_1} \le \frac{f(x_3)−f(x_2)}{x_3 −x_2}$
In $\mathbb{R}^n$ the points $x_1,x_2,x_3$ would be vectors, and we can't divide by a vector. But the following reformulation still makes sense:
If $x_2$ is between $x_1$ and $x_3$, then $$\frac{ f(x_2)−f(x_1)}{|x_2−x_1|} \le \frac{f(x_3)−f(x_2)}{|x_3 −x_2|}\tag1$$
where $|\cdot|$ is the Euclidean norm.
The proof is by reduction to one-dimensional cases. Let $g(t) = f(x_1+t(x_3-x_1))$. Then $g$ is convex on $\mathbb{R}$. Let $s = |x_2-x_1|/|x_3-x_1|$. This is a number between $0$ and $1$, so
$$\frac{ g(s)−g(0)}{s} \le \frac{g(1)−g(s)}{1-s}$$
Rewrite this back in terms of $f$ to get (1).