What to call the Euclidean norm divided by a constant

Question

I'm using the Euclidian distance $d_{2}$ divided by a constant $T$, i. e. $\frac{d_{2}}{T}$. However, I'm not sure what to call this. I'd like to keep things simple so I thought maybe "scaled Euclidian distance" but I'm not sure that is mathematically precise (enough).

Thanks!

Answer 1

Well you're using an equivalent Euclidian norm. Note the the definition of an norm equivalence is as follows:

$\| \cdot \|_1, \| \cdot \|_2$ are equivalent on some space if $\exists C_1,C_2 \in \mathbb{R}^+$ where $0 < C_1 \le C_2$ so that $\forall v$ in our space we have $$ C_1 \| v \|_2 \le \| v\|_1 \le C_2 \| v \|_2 $$

Now denote $\| \cdot \|_1$ to be the normal euclidian norm (i.e. the square root of the sum of the components squared) and $\| \cdot \|_2$ your norm. We have that for any $v$ $$ \| v \|_1 = T \| v \|_2 $$ so take $C_1 = C_2 = T$ to show the above inequality (as long as $T$ is a positive real number).

So in the end I would call your norm something like $\| \cdot \|_T$, an equivalent euclidian norm.