@YoavKallus That is also good try, thanks. The point in choosing small $|\gamma|$ is to ensure non negativity, right? This seems an interesting example disjoint from the class indicated by iosif Pinelis. Thanks you.

Marvellous idea, thanks a lot! I have checked without problems the (non-strict) convexity and the positive homogeneity of degree 1. I fail to see why they also satisfy point (3), would you mind expanding a bit your answer about this point? Thank you very much for your interest and for your answer!

@AlexandreEremenko I see your point and thanks for the comment, yet I am looking for a completely "different" example. I have not written it in the OP (to avoid being too long and verbose) but this function $f$ plays the role of integrand of a min problem in CoV, i.e. $\min \int_\Omega f(Du) dx$ among suitable competitors $u$. I am not interested into the case $f(\cdot) = \vert \cdot \vert$ (which is well-known) and, as you can now see, multiplicative constants do not play any role.

What a nice book the one by Torchinksy, I did not know it! It is a pity it does not have exercises on GMT. It reminds me somehow of the Costara-Popa (which is btw more on the functional analytic point of view). Thanks in any case for pointing it out!

@PiotrHajlasz Uh you are right, I thought it had problems also for multi-d case, but I got confused, you are right. Thanks!

Concerning Sobolev spaces, I would also recommend the well known book by Brezis, Functional Analysis, Springer.

Have you given a look to Functions of Bounded Variation and Free Discontinuity Problems (Luigi Ambrosio, Nicola Fusco, and Diego Pallara), Oxford Mathematical Monographs? The first two-three chapters may be of interest to you and they contain (non-trivial) exercises.

@πr8 Sure, I forgot to mention that I am looking for another function than the norm :-) You are perfectly right, thanks!