I have the following question. Suppose I have a random variable $X$ which takes on values in $S^{n-1}$ according to some distribution $\mu$. Now my goal is find another random variable $Y$ so that $\mathbb{E}[|X \cdot Y|]$ is maximal over all choices of $Y$ and $X$ and $Y$ are independent. My initial hunch is that choosing $Y \sim \mu$ maximizes $\mathbb{E}[|X\cdot Y|].$ However I'm not so convinced anymore. In particular I was thinking of the case where $X$ is uniformly distributed over the intersection of $S^{n-1}$ and the xy-plane and I take $Y := (0,0,1,0,...,0)$ and this seems like a reasonable choice for $Y$. I was wondering if anyone could direct me towards some math-literature related to this problem or if they could provide insight into this sort of problem.
Also this is my first post on the math.stackexchange so I apologize if my question isn't clear or broke a rule. I'll get the hang of things eventually.