# Geometric construction of copula - question regarding C-volume

I am learning about copula's, using Nelsen's book, and more specifically about the geometric method of constructing copula's. The problem is replicated in the following link: http://www.stat.ubc.ca/lib/FCKuserfiles/file/huacopula.pdf (Slides 37 - 42).

My question (referring to the posted link above) is related to Slide 39, 40, and 41. In this problem, we have defined the support of the copula to be the triangle shown in the diagram on Slide 39,40, and 41. The line on the left can be shown to be $$u=a v$$ and the line on the right to be $$u = 1 - (1-a)v$$. From a general definition, the support of a function is the subset of the domain of the copula where the value of the copula is not 0. So now onto my questions:

1. In Slide 39, $$C_a(u,v) = V_{C_a}([0,u]\times [0,v])$$. Because $$V_{C_a}([0,u]\times [0,v]) = C(u,v) - C(u,0) - C(0,v) + C(0,0)$$ (definition of C-volume) and by the grounded property that $$C(u,0)= C(0,v)= C(0,0)=0$$, we get that $$V_{C_a}([0,u]\times [0,v]) = C(u,v)$$ Because we are considering the region where $$u\leq av$$ (i.e. u is restricted by the left line segment, v can go from 0 to 1), the C-volume calculation becomes $$V_{C_a}([0,u]\times [0,1]) => C(u,v) = u$$ Is my reasoning sound here (mainly the part where I say u is restricted by the line and v can go from 0 to 1)?
2. In Slide 40, we consider the region where $$1−1−avuav$$. This equation is clear to me, as it is the region bounded by the equations I gave above for the support of the copula. However, Nelsen's reasoning for saying that $$C_a(u,v) = C_a(av,v) = av$$ is that the C-volume of of the rectangle $$[av,u]\times[0,v]$$ is 0. This is the confusing part to me. I understand that we defined the support of the copula, so that leads me to think that every term in the calculation of the C-volume will be 0, because the rectangles in this region do not fall on the support of the copula. But this seems like circular logic to justify the statement that the C-volume of the rectangle is 0. More specifically, it is my understanding that Nelsen arrives at the answer as follows: $$V_{C_a}([av,u]\times[0,v]) = C(u,v)-C(av,v)-C(u,0)+C(av,0) = 0$$ which implies $$C(u,v) = C(av,v)$$ Then using the result from Slide 39, we make the leap that $$C(u,v) = C(av,v) = av$$ However, my lack of understanding in this is as follows: no points (u,v) as defined in this region will be on the support of the copula, and as such the conclusion that I am wanting to draw (and most likely incorrectly) is that $$C(u,v)=C(av,v)=C(u,0)=C(av,0)=0$$ So my question is, why is this incorrect?
3. In proving slide 41, to me it seems like the same logic is applied as in question 2. Again, what am I missing?

I believe this question is asking how the copula $$C$$ can be found when the specified support is known to be as in the diagram below.
Thus, for all three cases, we can find the copula $$C(u,v)$$ as follows,
Hence, the obtained copula is written as $$C(u,v)=\begin{cases} u & \text{if u<\alpha v} \\ \alpha v & \text{if u>\alpha v and u<1-(1-\alpha )v} \\ u+v-1 & \text{if u>1-(1-\alpha)v} \end{cases}$$