I have two vectors $x, y \in \mathbb{R}^d$, it is well known as Minkowski inequality that:
$|x+y|_\infty \leq |x|_\infty + |y|_\infty$,
where $|x|_\infty= \underset{i=1..d}{\max} |x_i|$ with $x = \left[x_1 \dots x_d\right]^T$.
Could anyone kindly tell me when this inequality becomes equality?
I see at least that the equality happens if $x = \lambda y$ with $\lambda \geq 0$, but i have not found any other case (i.e. $x = Ay$ with $A \in \mathbb{R}^{d \times d}$?) after some hours looking for on the internet.
Thank you for your time!