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In this paper we prove that if (u, K) is an almost-minimizer of the Griffithfunctional and K is $\epsilon$-close to a plane in some ball B $\subset$ R Nwhile separating the ball B in two big parts, then K is C 1,$\alpha$ in aslightly smaller ball. Our result contains and generalizes the 2 dimensionalresult of [4], with a different and more sophisticate approach inspired by [23,24], using also [20] in order to adapt a part of the argument to Griffithminimizers.

Epsilon-regularity for Griffith almost-minimizers in any dimension under a separating condition