*Published Paper*

**Inserted:** 22 feb 2000

**Last Updated:** 19 dec 2001

**Journal:** Advances in Mathematics

**Number:** 159

**Pages:** 51-67

**Year:** 2001

**Abstract:**

We prove that for any set of finite perimeter in an Ahlfors $k$-regular metric space admitting a weak $(1,1)$-Poincaré inequality the perimeter measure is concentrated on the essential boundary of the set, i.e. the set of points where neither $E$ nor its complement have zero density. Moreover, the perimeter measure is absolutely continuous with respect to the Hausdorff measure $\cal H^{k-1}$. We obtain also density lower bounds on volume and perimeter which lead to the (asymptotic) doubling property of the perimeter measure: this property is of interest in connection with the rectifiability problem in the Heisenberg group and, more generally, in Carnot-Carath{é}odory groups.

**Keywords:**
perimeter, metric spaces

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