I am working on concave functions in $\mathbb{R}^n$ and one of the main results is a "reduction" one which allows to reduce the study of the concavity of a function of several variables to that of one variable. For this, we need to consider a convex subset $C\subset\mathbb{R}^n$ and a line in $\mathbb{R}^n$ that is a one dimensional subspace : $ A = \{ X\in\mathbb{R}^n : X= tV, t\in\mathbb{R} \}$. In my textbook, the intersection is represented by the following subset in $\mathbb{R}$ :
$$ \forall (X,V)\in C\times\mathbb{R}^n : J(X, V) = \{ t\in\mathbb{R} \mid X + tV \in C\} $$
My questions is the following : how can we see that it is indeed the intersection of $C$ with a line ? I mean, when I take $\mathbb{R}^3$ to have a picture I see that this intersection is also a line so it is a one dimensional space, however this intersection should be composed of vectors in $\mathbb{R}^n$ whereas the set $J(X,V)$ is in $\mathbb{R}$. Moreover, I don't see where the condition $X+ tV\in C$ comes from ?
Thinking on that, I thought the set $J(X,V)$ is a way to identify each vector in $\mathbb{R}^n$ that is both in $C$ and in $A$ since it is necessarly of the form $tV$... but I did not managed to find why the condition $X+ tV\in C$, is it something like if $X\in C$ and $X+tV\in C$ it implies that $tV\in C$ ? But I am not able to see why it should be true in general if we don't have a vector subspace structure (stable by the addition of a vector).
Please let me know where I am wrong and thank you a lot.