Let $K \subset \mathbb{R}^n$ denote a convex body. Let $\Pi_K$ denote the projection onto $K$, $$ \Pi_K(y) = \mathrm{arg\,min}_{x \in K} \|y - x\|, $$ where $\|\cdot\|$ denotes the usual Euclidean norm.
Suppose that $g \in \mathbb{R}^n$ is a random, standard Gaussian vector. I am wondering if there is a sharp characterization of the quantity $$ \mathbb{E}\Big[\|\Pi_K(g)\|^2\Big], $$ in terms of purely geometric/metric quantities relating to $K$. Of course, this can be further written as $$ \mathbb{E}\Big[\|\Pi_K(g)\|^2\Big] = \|\mu\|^2 + \mathbb{E} \Big[\|\Pi_K(g) - \mu\|^2\Big]$$ where $\mu = \mathbb{E} \Pi_K(g)$. I would also be interested to know what is known about $\mu$.
