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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.

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

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  • $\begingroup$ 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 $\lambda$ $\endgroup$ – DimSum Nov 22 '19 at 13:34
  • $\begingroup$ Then you should check your calculations. It should be the same $\lambda$. $\endgroup$ – gerw Nov 22 '19 at 20:51
  • $\begingroup$ Yes, I made a mistake, thanks I have it. $\endgroup$ – DimSum Nov 23 '19 at 9:45

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