# understanding an estimation of the perimeter of sets, $|P(\hat{V})-P(V)|\le P(B_r(x))$

Let $$V$$ be a Borel set in $$\mathbb{R}^n$$ such that the Lebesgue measure of $$V$$, $$|V|$$, satisfies $$|V|\approx |B_1(0)|$$, but $$|V|\neq |B_1(0)|$$ (i.e. $$|V|$$ is slightly greater or less than $$|B_1(0)|$$).

We modify the set $$V$$ to obtain a set $$\hat{V}$$ satisfying $$|\hat{V}|=|B_1(0)|$$ in the following way:

Fix a point $$x$$ on the boundary of $$V$$, $$\partial V$$, and let $$r>0$$ such that if we set $$\hat{V}:=\begin{cases} V\cup B_r(x),\;\text{if}\; |V|\le |B_1(0)|\\ V\setminus B_r(x),\;\text{if}\;\; |V|> |B_1(0)|\\ \end{cases}$$ we have $$|\hat{V}|=|B_1(0)|$$.

Let $$P(E)=\sup\left\{\int_{\mathbb{R}^n}\chi_E(x) \mathrm{div}\boldsymbol{\phi}(x) \, \mathrm{d}x : \boldsymbol{\phi}\in C_c^1(\mathbb{R}^n,\mathbb{R}^n),\ \|\boldsymbol{\phi}\|_{L^\infty(\mathbb{R}^n)}\le 1\right\}$$ is be the perimeter of a borel set $$E$$. See https://en.wikipedia.org/wiki/Caccioppoli_set

Why is $$|P(\hat{V})-P(V)|\le P(B_r(x))$$?

Heuristically, if we draw a picture, the estimate is clear, but I don't know how to prove it analytically. I tried to prove it with a case distinction, case $$|V|\le |B_1(0)|$$ and case $$|V|> |B_1(0)|$$ and using that we have the following for the characteristic functions: $$\chi_{V\cup B_r(x)}=\chi_V+\chi_{B_r(x)}-\chi_{V\cap B_r(x)}$$ and $$\chi_{V\setminus B_r(x)}=\chi_V-\chi_{V\cap B_r(x)}$$. However, I am not sure how to handle |sup.. minus sup..|calculations.

The estimate is false. Suppose $$B$$ is a ball with radius slightly smaller (or slightly larger) than 1, choose point $$x$$ on the boundary. Now you can modify your set on a small neighbourhood of $$x$$ so that the Lebesgue measure doesn't change much, but the perimeter increases arbitrarily (think of a comb-like shape on the boundary). That's your $$V$$, with $$|V|\not=|B_1(0)|$$. Now, the perimeter of $$\hat{V}$$ will be small again, as long as $$B_r(x)$$ contains the "comb" we created earlier. Then $$|P(\hat{V})-P(V)| \gg P(B_r(x)),$$ with the left hand side arbitrarily large.
In general, for $$E$$ and $$F$$ sets of finite perimeter, you have $$P(E\cup F) + P(A\cap F) \le P(E)+P(F),$$ and $$P(E\setminus F)\le P(E)+P(F),$$ see the book of Maggi on sets of finite perimeter, (12.17) and (12.19).