Lempert proved that the Kobayashi metric and the Carathéodory metric of smooth strongly convex domains are smooth away from the zero section of the holomorphic tangent bundle. We will show that if the strong convexity is replaced by the weaker condition of strict convexity the above smoothness result is no longer valid even if the boundary is real analytic. We study the smoothness of the Kobayashi metric of ellipsoids. We prove that the Kobayashi metric of domains of the form, wherem≥3/2, is piecewiseC3off the zero section, but notC3.