Power of Absolute Value is Convex Real Function

Theorem

Let $p \ge 1$ be a real number.

Define $f : \R \to \R$ by:

$\map f x = {\size x}^p$

for each $x \in \R$.

Then $f$ is a convex function.

Proof

From Absolute Value Function is Convex:

$x \mapsto \size x$ is a convex function.

Note now that:

$x \mapsto x^p$ is increasing on $\hointr 0 \infty$

From Power Function is Convex Real Function, we also have:

$x \mapsto x^p$ is convex.

Since $f$ is the composition of the maps $x \mapsto \size x$ and $x \mapsto x^p$, we have:

$f$ is convex

by Convex Real Function Composed with Increasing Convex Real Function is Convex.

$\blacksquare$