Intersecting a convex set with a ball exactly at two points

In the following I am using the usual Euclidean norm. Let $W\subset\mathcal{R}^2$ be a non-empty bounded convex set. Let $y_0\in \partial W$ (boundary of $W$). I want to argue: there exists a $\delta>0$ such that $\forall 0<\delta'<\delta$, $B_{\delta'}(y_0)$ intersects with $\partial W$ exactly at two points. Is the statement true? If so, how can I make the argument rigorous? If not, could you please provide a counter example? By drawing diagrams in $\mathcal{R}^2$, my intuition tells me it should be true. Any help would be greatly appreciated.