Convex loss function & normal posterior - Bayes's rule?

[Boot20]

Homework Statement

$W(\theta - \delta)$ the loss function.
$\theta$ the true parameter.
$\delta$ an estimator of $\theta$
W a smooth, non-negative, symmetric, convex function.
$p(\theta | x)$ the posterior density of the parameter $\theta$.

Prove that, for normal posterior density $p(\theta | x)$ and under the loss function specified, the estimator that minimize the expected loss is the posterior mean.

Homework Equations

The expected loss can be written as
$U(\delta) = E[W(\theta - \delta)] = \int W(\theta - \delta) p(\theta | x)d\theta$

The Attempt at a Solution

I really have no idea where to start. I suspect that $U(\delta)$ is also convex but I know no way to prove it.

 

[kurt.physics]

I don't think I can use the fact that W is convex to prove this as W is a function of \theta - \delta. Any help will be greatly appreciated.