Uniqueness and shape of simplex ground states for aggregation--diffusion energies with mild repulsion

We study global minimizers in Rᵈ of the free energy E_κ=W+κU, where W is a power-law interaction energy with mild repulsion and U is the porous-medium internal energy with exponent $m>1$. In a range of exponents that we call the strict simplex regime, results of Lim--McCann and Davies--Lim--McCann show that W is minimized, uniquely up to rigid motions, by equal point masses at the vertices of a regular unit d-simplex. Diffusion excludes such atomic minimizers for every $κ>0$. We prove that uniqueness nevertheless persists: for every fixed $m>1$ and all sufficiently small $κ>0$, the free energy has a unique global minimizer up to rigid motions. After a suitable rigid motion, this minimizer is invariant under every symmetry of the simplex, and its support splits into $d+1$ clusters near the vertices, each of mass exactly $1/(d+1)$; in particular, it is not radially symmetric when d≥2. The clusters have width of order κ1/(d(m-1)+2), which is also the exact order of the quadratic Wasserstein distance from the minimizer to the atomic simplex measure. Rescaled at this scale, the clusters converge in L¹ and L^∞ to explicit Barenblatt-type profiles supported on ellipsoids determined by the Hessian of the simplex potential. For d≥2 these ellipsoids are spheroids flattened along the directions of the vertices; when the repulsive exponent exceeds two, their aspect ratio equals (d+1)-1/2, independently of all other parameters. At the endpoint $(4,2)$ of this regime in dimensions d≥2, where the zero-diffusion minimizer is not unique, we show instead that vanishing diffusion selects the uniform spherical shell.