Given a convex set $K$ in Euclidean space and a point $O\in\partial K$, there exists an $\epsilon>0$ such that for all $r<\epsilon$ the intersection $S(O,r)\cap K$ is connected.
Thus, let $O\in \partial K$. Call $P\in \partial K$ a critical point if $\langle O-P, X-P\rangle \geq 0$ for all $X\in K$. Note that if $P_1$ and $P_2$ are critical points with $\angle P_1 O P_2 \leq \frac{\pi}{3}$ then by the Pythagorean theorem $\frac{|OP_1|}{|OP_2|}\leq 2$. Therefore by a simple packing argument $$\epsilon=\inf_{P \; critical} |OP|>0.$$ For every $C>0$ smaller than this infimum, we show that the sphere $S_C$ of radius $C$ centered at $O$ has the property that $S_C\cap K$ is connected. Indeed, if points $X,Y$ were in different connected components of $S_C\cap K$, we would connect them by a path in $K$ and then "push out" the path away from $O$ using a flow in $K$, using that fact that the flow can only get stuck at a saddle point of $\partial K$ for the distance function, and a saddle point is necessarily a critical point. The construction of the flow in the absence of such Grove-Shiohama critical points was described in http://link.springer.com/article/10.1007/BF02187719
For example, for an acute triangle in the plane, the optimal $\epsilon$ for a point on one of the sides will be the smaller of the two distances from $O$ to the remaining two sides. Every circle of radius smaller than $\epsilon$ will meet the triangle in a connected arc.