We establish various extensions of the comonotone improvement result of Landsberger and
Meilijson [Landsberger, M., Meilijson, I., 1994. Co-monotone allocations, Bickel–Lehmann
dispersion and the Arrow–Pratt measure of risk aversion. Annals of Operations Research 52,
97–106] which are of interest for the risk sharing problem. As a consequence we obtain
general results of the comonotonicity of Pareto optimal risk allocations using risk measures
consistent with the stochastic convex order.