In this paper, we describe the structure of shape derivatives around sets
which are only assumed to be of finite perimeter in RN. This structure
allows us to define a useful notion of positivity of the shape derivative and
we show it implies its continuity with respect to the uniform norm when the
boundary is Lipschitz (this restriction is essentially optimal). We apply this
idea to various cases including the perimeter-type functionals for convex and
pseudo-convex shapes or the Dirichlet energy of an open set