Absolute Value Function is Convex/Proof 2

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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 $\alpha + \beta = 1$.

\(\displaystyle \map f {\alpha x + \beta y}\) \(=\) \(\displaystyle \size {\alpha x + \beta y}\) Definition of $f$
\(\displaystyle \) \(\le\) \(\displaystyle \size {\alpha x} + \size {\beta y}\) Triangle Inequality for Real Numbers
\(\displaystyle \) \(=\) \(\displaystyle \size \alpha \size x + \size \beta \size y\) Absolute Value Function is Completely Multiplicative
\(\displaystyle \) \(=\) \(\displaystyle \alpha \size x + \beta \size y\) Definition of Absolute Value
\(\displaystyle \) \(=\) \(\displaystyle \alpha \, \map f x + \beta \, \map f y\) Definition of $f$

Hence the result by definition of convex real function.

$\blacksquare$