For multivariate normal posterior with improper prior, why posterior is proper only if $n\geq d$ This is related to Gelman's BDA chapter 3 section 5's noninformative prior density for $\mu$.
Let $\Sigma$ be fixed positive definite symmetric matrix of size $d$ by $d$. Let $y_1,\dots, y_n$ be iid Gaussian r.v. with density normal $N(\mu,\Sigma)$. Let $p(\mu)$ be prior density of $\mu$ which is proportional to constant. Then posterior $p(\mu|y_1,\dots, y_n)$ is proportional to $p(y_1,\dots, y_n|\mu)$. In particular, $p(\mu|y_1,\dots, y_n)$ is proportional to $|\Sigma|^{\frac{n}{2}}\exp(-\frac{1}{2}tr(\Sigma^{-1}S_0))$ (3.11) where $S_0=\sum_i(y_i-\mu)(y_i-\mu)^T$.
The book says (3.11) is a proper posterior only if $n\geq d$ and if $n<d$, then $S_0$ is not of full rank.
Q1. Why proper posterior only if $n\geq d$ here? I have tried $n=1,d=2$ and I still find the posterior proper. The essential thing I see is that $tr(\Sigma^{-1}S_0)$ for this case's quadratic form remains to be $\Sigma^{-1}$ with $(\mu_1-y_{11},\mu_2-y_{12})$ where $y_1=(y_{11},y_{12})$.
Q2. What does it mean for $S_0$ is not of full rank for $n<d$? Consider points $y_1=y_2=\dots=y_n$. $S_0$ is of rank 1 in this case. Does the statement mean there is a dense open set of $(y_1,\dots, y_n)\in R^{nd}$ s.t. $rk(S_0)$ is full for $n\geq d$.

Q¹: With a flat prior, and one single observation $x$,
$$\pi(\mu|x)\propto\exp\{-(x-\mu)^\mathsf{T}\Sigma^{-1}(x-\mu)\big/2\}$$
which implies that the posterior is a multivariate $\mathsf{N}_d(x,\Sigma)$, definitely proper. Extends straightforwardly to more than one observation.
Q²: When $n=d$, $$S_0=\sum_{i=1}^d(y_i-\mu)(y_i-\mu)^\mathsf T$$is not of full rank iff there exists a non-zero $\varepsilon$ such that $S_0\varepsilon=0$, i.e.,$$\sum_{i=1}^d(y_i-\mu)(y_i-\mu)^\mathsf T\epsilon=\sum_{i=1}^d\{(y_i-\mu)^\mathsf T\epsilon\}(y_i-\mu)=0$$implying that (i) $\epsilon$ is orthogonal to all $y_i-\mu$'s which thus (ii) live in a subspace of dimension $d-1$ at most, a zero measure set. Extends straightforwardly to $n>d$.
Notice. While the 2014 edition does contain this$-$unnecessary when $\Sigma$ is known$-$ condition, the latest printing has removed it:

