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In this paper, we establish a $C^{1,\alpha}$-regularity theorem for
almost-minimizers of the functional $\mathcal{F}_{\varepsilon,\gamma}=P-\gamma P_{\varepsilon}$, where $\gamma\in(0,1)$ and $P_{\varepsilon}$ is a nonlocal
energy converging to the perimeter as $\varepsilon$ vanishes. Our theorem
provides a criterion for $C^{1,\alpha}$-regularity at a point of the boundary
which is uniform as the parameter $\varepsilon$ goes to $0$. As a consequence
we obtain that volume-constrained minimizers of
$\mathcal{F}_{\varepsilon,\gamma}$ are balls for any $\varepsilon$ small
enough. For small $\varepsilon$, this minimization problem corresponds to the
large mass regime for a Gamow-type problem where the nonlocal repulsive term is
given by an integrable kernel $G$ with sufficiently fast decay at infinity.

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