# Convexity and order of points

Let $$x_1,x_2,y_1,y_2$$ be scalar points such that $$x_1 and $$y_1. 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.

Hint: Represent both $$x_1 - y_1$$ and $$x_2 - y_2$$ as convex combinations of the points $$x_1 - y_2$$ and $$x_2 - y_1$$.
• I can see that if if have $x:=x_1 - y_2$, $y := x_2 - y_1$ and $\lambda \in (0,1)$ such that $h(x_1-y_1) = h(\lambda x + (1-\lambda )y) \leq \lambda h(x) + (1-\lambda)h(y)$ and $h(x_2-y_2) = h((1-\lambda ) x + \lambda y) \leq (1-\lambda) h(x) + \lambda h(y)$ then summing both terms give the results. But it is not possible to find such a $\lambda$ – DimSum Nov 22 '19 at 13:34
• Then you should check your calculations. It should be the same $\lambda$. – gerw Nov 22 '19 at 20:51