  M. Brion proved a convexity result for the moment map image of an irreducible
subvariety of a compact integral Kaehler manifold preserved by the
complexification of the Hamiltonian group action. V. Guillemin and R. Sjamaar
generalized this result to irreducible subvarieties preserved only by a Borel
subgroup. In another direction, L. O'Shea and R. Sjamaar proved a convexity
result for the moment map image of the submanifold fixed by an antisymplectic
involution. Analogous to Guillemin and Sjamaar's generalization of Brion's
theorem, in this paper we generalize O'Shea and Sjamaar's result, proving a
convexity theorem for the moment map image of the involution fixed set of an
irreducible subvariety preserved by a Borel subgroup.
