Let $T$ be a compact and symmetric subset of $\mathbb{R}^n$, i.e, $T=-T$. Let $g\sim \mathcal{N}(\mu, \sigma^2I_n)$ be a random Gaussian vector with a non-zero mean vector $\mu$.
My goal is to find a lower bound of $$A = \mathbb{E}(\max_{t\in T} g^\top t) - \max_{t\in T}\mu^\top t.$$ Note that an application of Jensen's inequality reveals that $A\ge 0$. Therefore, I am interested in finding a positive lower bound of $A$.
An upper bound of $A$ is relatively easier to obtain, as triangle inequality gives $A\le \mathbb{E}(\max_{t\in T}(g-\mu)^\top t)$, which can be further upper bounded by standard techniques using covering numbers for $\epsilon$-nets of $T$ and the maximal inequality for zero mean Gaussians.
However, when it comes to the lower bound, the only idea that I have is to use Sudakov type minoration. I have also heard about Talagrand's majoring measure approach, but I haven't used it. However, as per my knowledge, all these approaches are useful when the means are zero. For non-zero means, the $\max_{t\in T}\mu^\top t$ remains uncancelled, which is something I don't want.
Can somebody please throw some useful ideas on how to find a suitable lower bound, or maybe there is some method that I am not aware of? I will be really grateful if somebody can show me such a method. Thanks in advance.