Incorporating prior knowledge into image segmentation algorithms has proven useful for obtaining more accurate and plausible results. Two important constraints, containment and exclusion of regions, have gained attention in recent years mainly due to their descriptive power. In this paper, we augment the level set framework with the ability to handle these two intuitive geometric relationships, containment and exclusion, along with a distance constraint between boundaries of multi-region objects. Level set's important property of automatically handling topological changes of evolving contours/surfaces enables us to segment spatially-recurring objects (e.g., multiple instances of multi-region cells in a large microscopy image) while satisfying the two aforementioned constraints. In addition, the level set approach gives us a very simple and natural way to compute the distance between contours/surfaces and impose constraints on it. The downside, however, is a local optimization framework in which the final segmentation solution depends on the initialization. In fact, here, we sacrifice the optimizability (local instead of global solution) in exchange for lower space complexity (less memory usage) and faster runtime (especially for large microscopic images) as well as no grid artifacts. Nevertheless, the result from validating our method on several biomedical applications showed the utility and advantages of this augmented level set framework (even with rough initialization that is distant from the desired boundaries). We also compared our framework with its counterpart methods in the discrete domain and reported the pros and cons of each of these methods in terms of metrication error and efficiency in memory usage and runtime.