# 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 $$x \neq 0$$.