Euclidean norm of Kronecker product of vectors Suppose $x,y\in\mathbb{R}^2$ and let $\left\|\cdot\right\|$ denote the Euclidean norm. I am trying to find a bound for $\|x\otimes x - y\otimes y\|$ as a multiple of $\|x-y\|$. 
We have a special easy case when $y = 0$. In that case
$$
\|x\otimes x\|^2 = \sum_{i=1}^{2}\sum_{j=1}^{2}(x_ix_j)^2 = \left(\sum_{i=1}^{2}x_{i}^{2}\right)^2 = \|x\|^4.
$$
Can anyone help me with the case $y\neq0$?
 A: Note that
$$
\|x\otimes x - y \otimes y\|^2 = 
(x\otimes x - y \otimes y)^T(x\otimes x - y \otimes y) = \\
(x\otimes x)^T(x\otimes x) - 2 (x\otimes x)^T(y \otimes y) + (y \otimes y)^T(y \otimes y) = \\
x^Tx \otimes x^Tx - 2x^Ty \otimes x^Ty + y^Ty \otimes y^Ty =\\
\|x\|^4 - 2\langle x,y \rangle^2 + \|y\|^4
$$
Compare this to 
$$
\|x - y\|^2 = \|x\|^2 - 2 \langle x,y \rangle + \|y\|^2
$$

Suppose that $\|y\| = \alpha \|x\|$ for some $\alpha > 0$.  The ratio of these two quantities looks like
$$
\frac{\|x\|^4 - 2\langle x,y \rangle^2 + \|y\|^4}{\|x\|^2 - 2 \langle x,y \rangle + \|y\|^2} = \\
\frac{(1 + \alpha^4)\|x\|^4 - 2\beta^2}
{(1 + \alpha^2)\|x\|^2 - 2 \beta}, \qquad |\beta| < \alpha \|x\|^2 =\\
\alpha \|x\|^2 \frac{(1 + \frac 1{\alpha^4}) - 2\gamma^2}
{(1 + \frac 1{\alpha^2}) - 2 \gamma}, \qquad |\gamma| < 1
$$
It is clear that, with a fixed $\alpha = \frac{\|y\|}{\|x\|}$ and a fixed $\gamma = \frac{\langle x,y \rangle}{\|x\|\,\|y\|}$, this quantity is unbounded as $\|x\| \to \infty$.
