Verifying triangular inequality for $\big(\sqrt{|x|}+\sqrt{|y|}\big)^2$

Question

How is it possible to verify that the triangular inequality holds (or not) for the following:

$||x,y||^*:=\big(\sqrt{|x|}+\sqrt{|y|}\big)^2$ where $(x,y) \in \mathbb{R}^2$

Edit: I already found counter example to show the inequality is not satisfied and also founda a "graphical point of view" plotting the level sets for: $(\sqrt{|x+\tilde{x}|}+\sqrt{|y+\tilde{y}|})^2$ and for $(\sqrt{|x|}+\sqrt{|y|})^2+(\sqrt{|\tilde{x}|}+\sqrt{|\tilde{y}|})^2$ but I'd like to have a "direct computation".

Answer 1

Hint: Compute:

• $\bigl\lVert(0,0)-(1,0)\bigr\rVert^*$;
• $\bigl\lVert(1,0)-(1,1)\bigr\rVert^*$;
• $\bigl\lVert(0,0)-(1,1)\bigr\rVert^*$.