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 2If , then if and only if .

*Proof:* Basically follows by continuity of from the fact that for we have and for we have .

Lemma 3is 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 1Let ‘s take .The function fulfills the conditions we need and here are the level lines of the functional (which is

nota 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!