|Dept.||ShanghaiTech University, China|
We study the geometry of minimizers of an interaction energy, which is a Lyapunov functional for the aggregation equation. When the interaction potential is mildly repulsive, it is known to be hard to characterize those minimizers due to the fact that they break the rotational symmetry, suggesting that the problem is unlikely to yield to the usual convexity or symmetrization techniques from the calculus of variations. We prove that, if the repulsion is mild and the attraction is sufficiently strong, the minimizer is unique and exhibits a remarkable simplex-shape rigid structure. As the first crucial step we consider the maximum variance problem of probability measures under the constraint of bounded diameter, whose answer in one dimension was given by Popoviciu in 1935.