Let $g$ denote a standard Gaussian vector in $\mathbb{R}^n$, and $\|\cdot\|$ a norm.
Let $x \in \mathbb{R}^n$ and define $$ F(x) = \mathbb{E}[\|x + g\| - \|g\|]. $$ I am wondering if it is possible to have a sharp, up to multiplicative constants independent of dimension, characterization of this quantity.
For the case that $\|\cdot\| = \|\cdot\|_2$ is the Euclidean norm, it is possible to show that $$ F(x) = \Theta\Big(\, \min\Big\{\frac{\|x\|_2^2}{\sqrt{n}}, \|x\|_2\Big\}\Big) $$ where the $\Theta(\cdot)$ notation is only hiding universal constants.
Thus, my question is for a general norm. Is there an analogous characterization? One obvious fact, due to the triangle inequality, is that for $x \geq 2 \mathbb{E} \|g\|$, we have, $$ \|x\|/2 \leq F(x) \leq \|x\|. $$ Hence, the question is really for $x$ near zero (specifically $x \ll \mathbb{E} \|g\|$).
I suspect that for a general norm, it may require some type of geometric arguments.
