*Published Paper*

**Inserted:** 12 oct 1998

**Last Updated:** 10 dec 2013

**Journal:** Ricerche di Matematica

**Volume:** 48

**Number:** supplemento

**Pages:** 167-186

**Year:** 1999

**Abstract:**

Let $E\subset R^n$ be a quasi minimizer of perimeter, that is, a set such that $P(E,$ $B_\rho(x))\le (1+\omega(\rho))P(F,B_\rho(x))$ for all variations $F$ with $F\Delta E \subset\subset$ $B_\rho(x)$ and for a given function $\omega$ with $\lim_{\rho\to 0}\omega(\rho)=0$. We prove that, up to a closed set with dimension at most $n-8$, for all $\alpha<1$ the set $\partial E$ is an $(n-1)$-dimensional $C^{0,\alpha}$ manifold. This result is obtained combining the De Giorgi and Reifenberg regularity theories for area minimizers. Moreover we prove that, in the case $n=2$, $\partial E$ is a bi-lipschitz curve.

**Keywords:**
regularity, minimal surfaces, perimeter

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