end{aligned}We observe that this definition is the counterpart in the framework of sets of finite perimeter of the notion of quasi-minimum introduced by Giaquinta and Giusti in [79] in the context of variational integrals.
There exists a constant C(n) such that if E is a set of finite perimeter, with (|E|=|B_r|) for some (r>0), then begin{aligned} alpha (E ^2+D(E le Cr^{1-n}beta (E ^2.
Let E be a set of finite perimeter in (Omega ).
(Federer) Let E be a set of finite perimeter in (mathbb {R}^n).
Let E be a set of finite perimeter in (mathbb {R}^n).
Let E be a set of finite perimeter with finite measure and (kin {1,ldots,n-1}).
In the particular case that E is a set of finite perimeter with (|E| = |B|), setting (mu =frac{1}{omega _n}{chi _{_{E}}}dx) and (nu =frac{1}{omega _n}{chi _{_{B}}}dx), from the above theorem we may conclude (see also [97, Sect. 2.1]), that there exists a convex function (varphi :mathbb {R}^nrightarrow mathbb {R}) such that, setting (T:=nabla varphi ), then (T x in B) for a.e.
Then, there exists a subsequence (E_{h_k}) locally converging in measure in (Omega ) to a set E of finite perimeter in (Omega ).
If is -rectifiable, then it has locally finite perimeter in the sense of De Giorgi, and therefore a unit normal vector exists -almost everywhere on [7, Sections 3.2.14, 3.2.15].
In fact only a few years later Fuglede's result was extended to general sets of finite perimeter in (mathbb {R}^n) in two papers by Hall et al. [84] and by Hall [82].
To this aim, given a set of finite perimeter E and a ball (B_r y)) with the same volume as E, we are going to measure the distance from E to the ball in the following way, see Fig. 9.
