# Absolute Value Function is Convex/Proof 2

## Theorem

Let $f: \R \to \R$ be the absolute value function on the real numbers.

Then $f$ is convex.

## Proof

Let $x, y \in \R$.

Let $\alpha, \beta \in \R_{\ge 0}$ where $\alpha + \beta = 1$.

 $\ds \map f {\alpha x + \beta y}$ $=$ $\ds \size {\alpha x + \beta y}$ Definition of $f$ $\ds$ $\le$ $\ds \size {\alpha x} + \size {\beta y}$ Triangle Inequality for Real Numbers $\ds$ $=$ $\ds \size \alpha \size x + \size \beta \size y$ Absolute Value of Product $\ds$ $=$ $\ds \alpha \size x + \beta \size y$ Definition of Absolute Value $\ds$ $=$ $\ds \alpha \, \map f x + \beta \, \map f y$ Definition of $f$

Hence the result by definition of convex real function.

$\blacksquare$