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We consider a non-local interaction energy over bounded densities of fixedmass $m$. We prove that under certain regularity assumptions on the interactionkernel these energies admit minimizers given by characteristic functions ofsets when $m$ is sufficiently small (or even for every $m$, in particularcases). We show that these assumptions are satisfied by particular interactionkernels in power-law form, and give a certain characterization of minimizingsets. Finally, following a recent result of Davies, Lim and McCann, we givesufficient conditions on the interaction kernel so that the minimizer of theenergy over probability measures is given by Dirac masses concentrated on thevertices of a regular $(N+1)$-gon of side length 1 in $\mathbb{R}^N$.

On the existence of minimizing sets for a weakly-repulsive non-local energy