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

In this paper we prove that if (u, K) is an almost-minimizer of the Griffith functional and K is $\epsilon$-close to a plane in some ball B $\subset$ R N while separating the ball B in two big parts, then K is C 1,$\alpha$ in a slightly smaller ball. Our result contains and generalizes the 2 dimensional result 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 Griffith minimizers.