Let $W\subset \mathcal{R}^n$ be a bounded convex set. Let $y_0\in \partial \overline{W}$. I want to argue: there exists some $\delta>0$ such that $\forall 0<\delta'<\delta$, $B_{\delta'}(y_0)\setminus W$ is connected. Is the statement true? If so, how do I make the argument rigorous? If not, could you please provide a counter example?
From drawing little diagrams in $\mathcal{R}^2$, my intuition tells me it should be. Any help would be greatly appreciated.
Remark: A special case of this question, where $W$ is a ball has been asked and answered (here: Connectedness of complement of intersection of two balls ). I wonder the same argument would still work here. Any help would be greatly appreciated.