Provide an example where C is not closed and is not convex but distance function is still convex.

Question

I am looking for an example which shows that when the set C is not closed and C is not convex but dC= inf||x-c|| (distance function) can still be convex.

Answer 1

Delete the center from the (open or closed) unit disk.

Answer 2

Hint: The distance function of $C$ equals the distance function of its closure $\bar C$.