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

1. Is absolute value of convex function convex?
2. Is an absolute value function convex or concave?
3. What's the difference between norm and absolute value?

Is absolute value of convex function convex?

The absolute value function f(x)=|x| is convex (as reflected in the triangle inequality), even though it does not have a derivative at the point x=0.

Is an absolute value function convex or concave?

The real absolute value function is a piecewise linear, convex function.

What's the difference between norm and absolute value?

Norm is the generalized idea of distance. Absolute value is distance on the real line.