Let $X$ be a random vector, $x,y\in\mathbb{R}^{n}$ and $p\geq 1$. My question is the following. If we know that $\text{E}[\|X - x\|_{p}^{p}]\leq\text{E}[\|X - y\|_{p}^{p}]$ (provided that they exist), we can deduce that $\text{E}[\|X - x\|_{p}]\leq\text{E}[\|X - y\|_{p}]$ ?.
I try to use Jensen's inequality, but it doesn't work.
Thanks in advance.
Consider the case $n=1$, and $p=2$ (note that the statement is always trivially true for $p=1$).
We choose $x=0$, and $y=-1$, and define $X$ as a random variables on $\{-1,1\}$ such that $$ X = \begin{cases} 1 & \text{ with probability } 1/3\\ -1 & \text{ with probability } 2/3 \end{cases} $$ Clearly, $\mathbb{E}[(X-x)^2]=\mathbb{E}[|X-x|]=1$. However, one can easily compute that $$ \mathbb{E}[(X-y)^2] = \mathbb{E}[(X+1)^2] = \frac{4}{3} > \mathbb{E}[(X-x)^2] $$ but $$ \mathbb{E}[|X-y|] = \mathbb{E}[|X+1|] = \frac{2}{3} < \mathbb{E}[|X-x|] $$