  In an earlier paper \cite{mazeng} the authors introduced the notion of
curvature entropy, and proved
  the plane log-Minkowski inequality of curvature entropy under the symmetry
assumption. In this paper we demonstrate the plane log-Minkowski inequality of
curvature entropy for general convex bodies. The equivalence of the uniqueness
of cone-volume measure, the log-Minkowski inequality of volume, and the
log-Minkowski inequality of curvature entropy for general convex bodies in
$\mathbb R^{2}$ are shown.
