Consider two spherical Gaussian distributions in $\mathbb{R}^n$, $A = \mathcal{N}(x, I)$ and $B=\mathcal{N}(y, I)$ where the difference in means is $\delta = y - x$. Let $S \subset \mathbb{R}^n$ be a set with $\mathbb{P}[A \in S] = p$. I'm trying to show a lower bound on $\mathbb{P}[B \in S]$.

My intuition says that the lower bound is achieved when $S$ is a half-space with normal vector $\delta$. More specifically, I'm pretty sure the extremal $S$ is the half-space $S^* = \{z: \frac{\delta}{\| \delta\|} ^T z \le \Phi^{-1}(p) \}$, It's easy to verify that $\mathbb{P}(A \in S^*) = p$ and $\mathbb{P}(B \in S^*) = \Phi(\Phi^{-1}(p) + \|\delta\|)$.