Beautiful proof! Thank you.

I updated the proof of positive definiteness, the new one also gives an explicit bound.

You are right, perhaps that is not at all true. I do think we could find such $c_1$, $c_2$ though, I'll have to look around for a proof. We would also have to "normalize" away a constant since the proof obviously fails by considering the family of norms $\|x\| = a \|x\|_2$ as $a \to \infty$.

I was sloppy on the notation to save some space. To be more strict, the idea is that $X/Y$ are $\mathcal{X}/\mathcal{Y}$ valued random variables and hence $E(X|Y)$ is an $\mathcal{X}$ valued random variable which is a function of a $\mathcal{Y}$ valued random variable. Thus the function I'm looking for is thus more strictly $y \to E(X|Y=y)$.