# Is Sum of Absolute Value / ${L}_{1}$ Norm of Differences Convex?

I'm not sure how to approach this exercise.

One idea is to derive it w.r.t z, show that there is a min-extremum at $$z=f_k$$ and then show that for each value from the right and the left of the loss function it is positive which will prove that it is convex.

I never worked with Dirichlet functions, I'm not sure how should I show that it is convex analytically. Can somebody give me a clue/solution path ?

• You didn't include an actual question... – MBaz Nov 23 '20 at 14:53
• fixed it, thanks. – Ilya.K. Nov 23 '20 at 14:56

The 1st atom is the Absolute Value function $$\left| \cdot \right|$$ which is convex.