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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 ?

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    $\begingroup$ You didn't include an actual question... $\endgroup$ – MBaz Nov 23 '20 at 14:53
  • $\begingroup$ fixed it, thanks. $\endgroup$ – Ilya.K. Nov 23 '20 at 14:56
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It is easier to prove it by using atoms.
The 1st atom is the Absolute Value function $ \left| \cdot \right| $ which is convex.
Then you have linear operation by the subtraction which is convex (Also concave).
Then you linear combination which is also Convex.

Hence the function is Convex.

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