# How can I show convexity of this risk function?

I have the following risk function:

$$\mathbf{Risk}(x):=\mathbb{E}[R(x)]+\delta\mathbb{E}[|R(x)-\mathbb{E}[R(x)]|]$$

where $$R(x)$$ is the portfolio return and $$\delta$$ is any positive scalar. My textbook assumes this function is convex without proving it. Any ideas how it could be shown it is convex?

Since $$R(x)$$ is linear in $$x$$, we just need to prove $$\mathbb{E}(x) + \delta \mathbb{E}(\mathbb{|x-\mathbb{E}(x)|})$$ is convex. Now $$\mathbb{E}(x)$$ is linear and $$\delta$$ is a positive constant, so we only need to prove $$f(x):=\mathbb{E}(|x-\mathbb{E}(x)|)$$ is convex in $$x$$. Then it is straightforward to prove by definition: for $$0\leq t <1$$,
$$f(tx+(1-t)y) = \mathbb{E}(|tx + (1-t)y -\mathbb{E}(tx + (1-t)y)|) = \mathbb{E}(|t(x-\mathbb{E}(x)) + (1-t)(y-\mathbb{E}(y))|) \leq \mathbb{E}(|t(x-\mathbb{E}(x))| + |((1-t)(y-\mathbb{E}(y))|) = tf(x) + (1-t)f(y)$$,