Positive lower bounds for $|x+y|^2 -|x|^2$

Question

Let $x, y\in\mathbb{R}^3$ two nonnull vectors. I am looking for some positive lower bounds for $$|x+y|^2 -|x|^2,$$

where $|\cdot|$ stand for the magnitude of the vector.

$\textbf{My attempt:}$ By using the definition, I got $$(x_1 +y_1)^2 +(x_2 +y_2)^2+(x_3 +y_3)^2 -x_1^2 -x_2^2 -x_3^2 = |y|^2 +2x_1 y_1 +2 x_2 y_2 +2x_3 y_3 = |y|^2 +2\sum_i x_iy_i $$ but I think it is not so much useful to find some lower bounds.

Could someone please help?

Thank you in advance.

$\textbf{EDIT:}$ I missed to ask if there exists a positive lower bound for the similar case $$|x|^2 -|x-y|^2.$$

If someone would answer (or edit its answer), it would be great. If not, I will post a new question.

Answer 1

Whatever norm you chose on $\Bbb R^3$ (or on any real vector space) $$\forall x\ne0\quad\min_{y\ne0}\left(\|x+y\|-\|x\|\right)=-\|x\|$$ (attained for $y=-x$) hence $$\inf_{x,y\ne0}\left(\|x+y\|-\|x\|\right)=\inf_{x\ne0}\left(-\|x\|\right)=-\infty.$$

Answer 2

There is no positive lower bound, you can make both $|y|^2$ and $2\langle x,y \rangle$ as small as you want. Infact there is no non-positive lower bound either, since $\langle x,y \rangle$ can also be made as negative as you want.