If we have a $n$ dimensional vector $x$ and we create a copy of this vector and denote it $y$. Suppose we add another dimension with a of scalar $h>0$ to $y$, so that $y$ is an $n+1$-dimensional vector and the final entry of $y$ is $h$. Is there any way we can write the following in terms of $h$?
$$\|x\|_2 - \|y\|_2$$
I'm aware of the triangle inequality but I am looking for an exact value or a tighter bound. Further, can we generalise this beyond adding a single dimension, and think about adding $m\leq n$ dimensions with the same scalar $h$?
Here's a quick coding example to showcase what I mean
x=c(rnorm(3,mean=2))
h=0.1
y = c(x,h)
norm(x,type="2") - norm(y,type="2")