# Least area bounded by multiple of curves

In this paper SOME EXTREMAL QUESTIONS FOR SIMPLICIAL COMPLEXES, the author discussed about minimal area of bounded by multiples of a curve. Say we have a (well-behaved) curve $$\Gamma$$, the solution to the minimal area problem is $$S$$. Now e consider $$2\Gamma$$, he solution to the minimal area problem is $$S'$$. But $$area(S')<2area(S)$$.

My question is what does $$2\Gamma$$ mean? Does it mean a similiar transformation in the coordinate multiplying by 2? If it is this case, then I cannot really imagine this is true.

As @alesia points out $$2\Gamma$$ means to take the curve with multiplicity two" (this is usually understood to be in the context of currents or flat chains -- for an oriented curve you can think about it as tracing out the curve twice). The reason this can give a lower area for the minimizer is that the space of competitors for $$2\Gamma$$ contains two times all competitors for $$\Gamma$$ but may also contain new surfaces.
If $$M$$ is a Mobius band in $$\mathbb{R}^3$$ and $$\Gamma=\partial M$$, then it is the case that $$2\Gamma=\partial \tilde{M}$$ where $$\tilde{M}$$ is the orientation double cover of $$M$$. If it is the case that $$M$$ is the least area surface bounded by $$\Gamma$$ (among all orientable and non-orientable surfaces), then one should expect that the least area of orientable surface bounded by $$\Gamma$$, $$N$$, (which exists by appealing to geometric measure theory results) can satisfy $$|N|>|M|$$ (here $$|N|$$ and $$|M|$$ are the areas of $$N$$ and $$M$$, respectively). However, the least area orientable surface bounded by $$2\Gamma$$ should have area at most $$2|M|$$ (since $$\tilde{M}$$ is a valid competitor) and so one has $$|N'|\leq |\tilde{M}|= 2|M|<2|N|$$ where $$N'$$ is the least area orientable surface spanning $$2\Gamma$$.
I do not have access to the paper, but in this literature multiplying a curve by say $$2$$ roughly means taking the union of the curve with a very close translate of that curve. In reality, the translate actually coincides with the original curve, so we have "twice the same curve". This can be given a precise meaning in the formalism of currents.