Let $A$ be a $C^*$-algebra acting on a Hilbert space that admits a cyclic unit vector $\xi \in H$. Pose $S_\xi = \{\eta \in A \xi: \| \eta \| = 1\}$, and for each $\eta \in S_\xi$, pose $A_\eta = \{a \in A : a \xi = \eta\}$, so $A_\eta \ne \emptyset$ by definition; and let $c_\eta = \inf\{\| a \| : a \in A_\eta\}$. Clearly, $c_\eta \ge 1$ for all $\eta \in S_\xi$. I am interested in the following questions.

**Question 1**. Is it true that $c_\eta = 1$ for all $\eta \in S_\xi$ ?

**Question 1 (weak form)**. Is it true that $S(A)\xi$ is dense in $S(H)$, where $S(X)$ denotes the closed unit ball of a normed space $X$?

**Question 2**. If the answer of Question 1 turns out to be negative, can there be any examples for which $\sup\{c_\eta : \eta \in S_\xi\} = +\infty$ ?

**Question 3**. What if we drop the assumption of $\xi$ being cyclic but merely asks $A$ acts non-degenerately on $H$ ?