Convexity and order of points

Question

Let $x_1,x_2,y_1,y_2$ be scalar points such that $x_1<x_2$ and $y_1<y_2$. Let $h$ be a convex function from $\mathbb{R}$ to $\mathbb{R}_{+}$. Then,

$$h(x_1-y_1) + h(x_2-y_2) \leq h(x_1-y_2) + h(x_2-y_1)$$

I need this step to caracterise the solution of a 1D Monge Problem. It was thrown in a proof like if it is a trivial fact but I have no idea how to apply the convexity here.

Answer 1

Hint: Represent both $x_1 - y_1$ and $x_2 - y_2$ as convex combinations of the points $x_1 - y_2$ and $x_2 - y_1$.