Before revision: Equivalent characterizations of multi-portfolio time consistency are deduced for closed convex and coherent set-valued risk measures on L^p_d  . In the convex case, multi-portfolio time consistency is equivalent to a  condition on the sum of minimal penalty functions.
After revision: Equivalent characterizations of multi-portfolio time consistency are deduced for closed convex and coherent set-valued risk measures on L^p_d (F_T) with image space in the power set of L^p_d(F_t) . In the convex case, multi-portfolio time consistency is equivalent to a condition on the sum of minimal penalty functions.