$L_1$ distance between re-normalized points

Question

Let $\mathbf{x},\mathbf{y} \in (0, \infty)^d$. Are there general relations between $ \Vert \mathbf{x}-\mathbf{y} \Vert_1 $ and $ \left\Vert \frac{\mathbf{x}}{\Vert \mathbf{x}\Vert_1} -\frac{\mathbf{y}}{\Vert \mathbf{y}\Vert_1} \right\Vert_1 $? E.g. one is upper-bounded by a constant times the other etc.

Answer 1

Assume wlog $||x||_1\leq||y||_1$. We have \begin{equation} ||\frac{x}{||x||}-\frac{y}{||y||}||=\frac{1}{||x||}||x-\frac{||x||y}{||y||}||\\=\frac{1}{||x||}||x-y+(1-\frac{||x||}{||y||})y||\leq\frac{1}{||x||}||x-y||+(\frac{1}{||x||}-\frac{1}{||y||})||y||\\=\frac{1}{||x||}||x-y||+\frac{||y||}{||x||}-1 \end{equation}