  We consider monotone mean-variance (MMV) portfolio selection problems with a
conic convex constraint under diffusion models, and their counterpart problems
under mean-variance (MV) preferences. We obtain the precommitted optimal
strategies to both problems in closed form and find that they coincide, without
and with the presence of the conic constraint. This result generalizes the
equivalence between MMV and MV preferences from non-constrained cases to a
specific constrained case. A comparison analysis reveals that the orthogonality
property under the conic convex set is a key to ensuring the equivalence
result.
