I'm trying to find some operators $P:\mathbb{X} \to \mathbb{R}^n$ that satisfy these properties (it is okay to assume $\mathbb{X} \subset \mathbb{R}^n$ is compact):
(i) $\|\sum_{k=1}^N p_{i} P(x_i) - x \|_2^2 \le \|\sum_{k=1}^N p_{i} x_i - x \|_2^2$, where $x\in \mathbb{X}$, $x_i\in \mathbb{R}^n$, for $\sum_{k=1}^N p_{i}=1$.
(ii) For all $y \in \mathbb{X}$, $\|P(y)\|<B$ for some $B>0$.
I'm trying to find an operator $P$ that does not depend on $x$ (as otherwise I can set $P(y)=x$ for $y \in \mathbb{X}$). Would anybody have any ideas on how to do this? I've been struggling with this for quite a while and am wondering how to proceed. I've tried the projection operator and setting $\mathbb{X}$ to be a large ball, but no luck. I'm guessing a projection operator with $\mathbb{X}$ chosen to be a clever, convex-like set will work but have made no progress.