Pythagorean inequality in l1 norm for projection onto simplex

Question

Let $x$ be a point in $\mathbb{R}^d$, let $z$ be the projection of $x$ onto the $d-1$ dimensional simplex ($z^\top \mathbf{1}= 1$ and $z \succeq 0$). So basically

$$z = \arg\min_w \{\rVert x-w\rVert_2 \,\,|\,\, w^\top \mathbf{1} = 1\,,w\succeq 0 \} $$

Let $y$ be any other point in the simplex. Is it true that $\rVert z- y\rVert_1 \leq \rVert x- y\rVert_1$?

I could prove this for $\mathbb{R}^2$ but not sure how to generalize it to $\mathbb{R}^d$. The above inequality actually holds for Euclidean norm (not sure if it has an official name but some people refer to it as the Pythagorean inequality for projection onto convex sets)

but I'm interested in L1 norm instead. Any leads/related problems are greatly appreciated!

Thanks

Answer 1

It seems that the Pythagorean inequality only holds for inner-product norms rather than all norms. Thus this result may not hold for $\ell_1$ norm. One can refer to this book``Introduction to Online Convex Optimization'' (Page. 159) and this problem for more details.

Answer 2

Maybe I misunderstood the question, but it seems that even for $d=2$ the claim fails. Consider $x=(\frac{4.4}{9}, \frac{5.5}{9})$. Then $z=\frac{x}{1.1}=(\frac{4}{9}, \frac{5}{9})$. If we take $y=(\frac{3}{9}, \frac{6}{9})$, then $\|x-y\|_{1}=\frac{1.9}{9}$, while $\|z-y\|_{1}=\frac{2}{9}$.