  We consider finite-entropy solutions of scalar conservation laws $u_t +a(u)_x
=0$, that is, bounded weak solutions whose entropy productions are locally
finite Radon measures. Under the assumptions that the flux function $a$ is
strictly convex (with possibly degenerate convexity) and $a''$ forms a doubling
measure, we obtain a characterization of finite-entropy solutions in terms of
an optimal regularity estimate involving a cost function first used by Golse
and Perthame.
