A family of explicit minimizers for interaction energies

In this paper we consider the minimizers of the interaction energies with the power-law interaction potentials $W({\bf x}) = \frac{|{\bf x}|^a}{a} - \frac{|{\bf x}|^b}{b}$ in $d$ dimensions. For odd $d$ with $(a,b)=(3,2-d)$ and even $d$ with $(a,b)=(3,1-d)$, we give the explicit formula for the unique energy minimizer up to translation. For the odd dimensions, the key observation is that successive Laplacian of the Euler-Lagrange condition gives a local partial differential equation for the minimizer. For the even dimensions $d$, the minimizer is given as the projection and rescaling of the previously constructed minimizer in dimension $d+1$ via a new lemma on dimension reduction.