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