Let $C$ be an $n \times p$ matrix and $b$ be a column vector of length $n$, and $c>0$. Let $E := \{x \in \mathbb R^p \mid \|Cx-b\| \le c\}$, a hyperellipsoid in nonstandard position.
Question 1. What is a good analytic lower-bound for the quantity $\theta:=\underset{\|Cx-b\| \le c}{\inf} \|x\|^2$, say in terms of $c$, the singular-values of $C$, etc. ?
Note that this is the (squared) distance between E and the origin.
Question 2. Same question when $c=t\sqrt{n}$ with $t \in (0, 1)$, $C$ is a random matrix with rows independent iid rows from $N(0,1/p)$ and $b$ is a random vector with iid entries from $N(0,1)$, and independent of $C$. The question is then above high-probability lower-bounds for $\theta$.
Notes
The case when $b$ is the zero vector has been solved here https://math.stackexchange.com/a/2090988/168758. However, tbe general case seems to require a completely different argument / technique.
