A question dealing with the convexity of functions involving the absolute value

Question

Just beginning to learn convex analysis and optimization, I have some inquiries to make with regard to the absolute value function $f(x)= |x|$. This function is clearly convex, but since we know that $|x|= \max\{x,-x\}$, does this make $\max$ a convex function? How about $\min$? My guess is that the minimum function is concave, but I might be wrong. I'd appreciate any helpful input on these questions I'm having.

Answer 1

The optimisation result is that the pointwise supremum of affine functions is convex. On the contrary, the pointwise infimum of linear (or affine) functions is concave.

$\max\{x,-x\}$ is convex because it is the maximum of two linear functions of $x$, namely $x$ and $-x$.

Answer 2

You are right about the maximum, and you could find this fact in Wikipedia: properties of convex functions.

Also right about the minimum being concave. Remember that everything we say about convex functions translates into statements about concave functions simply by replacing $f$ with $-f$. This also switches $\max$ and $\min$ around.