Borsuk number for planar convex bodies

By using some simple tools from graph theory, we obtain a characterization of the planar convex bodies with Borsuk number equal to two. This result allows to give some examples of planar convex bodies with Borsuk number equal to three. Moreover, we also prove that the unique centrally symmetric planar convex bodies with Borsuk number equal to three are the Euclidean balls.