Integral as infimum of integrals I am trying to understand if the following formula holds. I cannot prove it but  cannot find a counterexample either.
For $\mu$ a probability measure on $\mathbb{R}^d$ and $p \geqslant 1$ does it hold that 
$\int |x|^p d\mu(x) = \inf \limits_{y \in \mathbb{R}^d } \int |x+y|^p d\mu(x)$ ?

Not always. For example, if $\mu$ is a Dirac measure at a point $x \neq 0$.
