A function is called -homogeneous, if for any it holds that . This implies that and thus, all -homogeneous functions on the positive reals are just multiples of powers of . If you have such a power of you can form the -norm
By Minkowski’s inequality, this in indeed a norm for .
If we have just some function that is not homogeneous, we could still try to do a similar thing and consider
It is easy to see that one needs and increasing and invertible to have any chance that this expression can be a norm. However, one usually does not get positive homogeneity of the expression, i.e. in general
A construction that helps in this situation is the Luxemburg-norm. The definition is as follows:
Definition 1 (and lemma). Let fulfill , be increasing and convex. Then we define the Luxemburg norm for as
Let’s check if this really is a norm. To do so we make the following observation:
Lemma 2 If , then if and only if .
Proof: Basically follows by continuity of from the fact that for we have and for we have .
Lemma 3 is a norm on .
Proof: For we easily see that (since ). Conversely, if , then but since this can only hold if . For positive homogeneity observe
For the triangle inequality let and (which implies that and ). Then it follows
and this implies that as desired.
As a simple exercise you can convince yourself that lead to .
Let us see how the Luxemburg norm looks for other functions.
Example 1 Let ‘s take .
The function fulfills the conditions we need and here are the level lines of the functional (which is not a norm):
[Levels are ]
The picture shows that this functional is not a norm ad the shape of the “norm-balls” changes with the size. In contrast to that, the level lines of the respective Luxemburg norm look like this:
[Levels are ]
June 17, 2020 at 11:28 am
Thank you for the very nice explanation!