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In this paper we study the blow-ups of the singular points in the boundary ofa minimizing cluster lying in the interface of more than two chambers. Weestablish a sharp lower bound for the perimeter density at those points and weprove that this bound is rigid, namely having the lowest possible densitycompletely characterizes the blow-up.

Lower bound for the perimeter density at singular points of a minimizing cluster in $\mathbb R^N$