Inequality euclidean norm

Question

I want to know if the following inequality holds: $(\sum\limits_{i=1}^{N}|x_i|-|y_i|)^{2}\le C\sum\limits_{i=1}^{N}(|x_i|-|y_i|)^{2}$ where $C>0$ is some constant.

Answer 1

Let's remind the Cauchy-Schwarz inequality in $\mathbb{R}^N$, for the standard inner product :

$$\forall a\in\mathbb{R}^N,\,\forall b\in\mathbb{R}^N,\,\left<a\mid b\right>^2=\Vert a\Vert^2\Vert b\Vert^2$$

that is, more explicitly :

$$\forall (a_1,\cdots,a_N)\in\mathbb{R}^N,\,\forall(b_1,\cdots,b_N)\in\mathbb{R}^N,\,\left(\sum_{i=1}^Na_ib_i\right)^2\le\left(\sum_{i=1}^Na_i^2\right)\left(\sum_{i=1}^Nb_i^2\right)$$

Now if we choose $a_i=\vert x_i\vert-\vert y_i\vert$ and $b_i=1$ for every $i\in\{1,\cdots,N\}$, then we get :

$$\left(\sum_{i=1}^N\vert x_i\vert-\vert y_i\vert\right)^2\le N\sum_{i=1}^N\left(\vert x_i\vert-\vert y_i\vert\right)^2$$