Interaction energy minimizers appear naturally in the study of collective dynamics in biological and social sciences. Despite the simple formulation, these minimizers may exhibit rich and unexpected pattern formation. The study of existence, uniqueness and qualitative behavior of minimizers is often challenging and requires the development of new theoretical tools. In this talk, I will discuss my recent results on the uniqueness of minimizers based on convexity theory, as well as the theoretical justification of annulus-shaped minimizers.