Given two intersecting sets of finite perimeter,E1 and E2,with unit normals v1 and v2 respectively,we obtain a bound on the integral of v1 over the reduced boundary of E1 inside E2.This bound depends only on the perimeter of E2.For any vector feld F: Rn→Rn with the property that F∈L∞ and divF is a(signed)Radon measure,we obtain bounds on the flux of F over the portion of the reduced boundary of E1 inside E2.These results are then applied to averaged shape optimization problems.