When is a $1$-varifold $V$ the associated varifold of the reduced boundary of some Caccioppoli set?

Question

Let $v_1$, $v_2$, $\cdots$, $v_l\in\mathbb{R}^n$ be unit vectors, $\mathbb{R}_v^+:=\{\lambda v:\lambda>0\}\subset\mathbb{R}^n$ be the ray in $v$'s direction; $n_1$, $n_2$, $\cdots$, $n_l>0$ be integers. I've proved that the $1$-varifold given by $$V=\sum_{j=i}^l n_j|\mathbb{R}_{v_j}^+|$$ is stationary if and only if the following balancing condition is satisfied in $\mathbb{R}^n$, $$\sum_{j=1}^ln_jv_j=\mathbf{0}.$$ Now my question is when $n=2$, what are the conditions on $v_j$, $n_j$ so that the $1$-varifold $V$ above is the associated varifold of the reduced boundary of some Caccioppoli set in $\mathbb{R}^2$?

Any comments and answers are appreciated!

Answer 1

I think the obvious way is indeed the only way. First of all $n_j=1$ for all $j$, as boundaries do not have higher multiplicity.

Secondly, the $l$ rays of your varifold split $\mathbb{R}^2$ into $l$ sectors (assuming the $v_i$ to be unique). Each sector is either completely included (up to the usual leftovers of measure zero) in you Caccioppoli-set or completely excluded, otherwise you would see a boundary somewhere in between. Similarly two adjacent sectors cannot both be included or excluded, otherwise there would be no reduced boundary in between. So if you have to alternate, than this means that $l$ has to be even.