Let $B\subseteq \mathbb{C}$ such that $$B:=\{z\in \mathbb{C}: |Re~z|\leq \varepsilon\},$$ for some fixed $\varepsilon \in [0,\frac{1}{2}]$. Let $A$ be a non-empty compact convex subset of $\mathbb{C}$. Then what is the form of the following set :
$$U:=\{\kappa\in S^1: \kappa A\cap B \neq \emptyset\},$$ where $S^1$ denote the unit circle in $\mathbb{C}$.
Edit: Here $\kappa A$ denotes the collection $\{\kappa a: a\in A\}.$
I strongly feel that $U=\pm D$ where $D$ is a compact connected subset of $S^1.$ Because, the convexity of $A$ assures us that "proccess of crossing the strip $B$" should be continuous and it cannot be discrete.
What I have tried so far: Consider any $a\in A$. Then obviously, we can find an unimodular constant $\mu$ which suitable raotates $a$ and puts it inside the strip $B$. Therefore, $U$ is clearly non-empty. Now, consider any $a_1,a_2\in A$ such that there exist $\mu_1,\mu_2\in S^1$ with $|Re~\mu_1a_1|\leq \varepsilon$ and $|Re~\mu_2a_2|\leq \varepsilon$. What I am trying to show is that for all $\gamma$ either of the form $$\gamma= \frac{t\mu_1+(1-t)\mu_2}{|t\mu_1+(1-t)\mu_2|}$$ or of the form $$\gamma= \frac{t\mu_1-(1-t)\mu_2}{|t\mu_1-(1-t)\mu_2|}$$ there exists at least one $a_\gamma\in A$ such that $|Re~\gamma a_\gamma| \leq \varepsilon.$ Here, I got stuck.