Multiphase active contour based models are useful in identifying multiple regions with spatial consistency but varying characteristics such as the mean intensities of regions. Segmenting brain magnetic resonance images (MRIs) using a multiphase approach is useful to differentiate white and gray matter tissue for anatomical, functional and disease studies. Multiphase active contour methods are superior to other approaches due to their topological flexibility, accurate boundaries, robustness to image variations and adaptive energy functionals. Globally convex methods are furthermore initialization independent. We extend the relaxed globally convex Chan and Vese two-phase piecewise constant energy minimization formulation to the multiphase domain and prove the existence of a global minimizer. An efficient dual minimization implementation of our binary partitioning function model accurately describes disjoint regions using stable segmentations by avoiding local minima solutions. Experimental results indicate that the proposed approach provides consistently better accuracy than other related multiphase active contour algorithms using four different error metrics (Dice, Rand Index, Global Consistency Error and Variation of Information) even under severe noise, intensity inhomogeneities, and partial volume effects in MRI imagery.