This is a problem from financial mathematics. I have never studied convex analysis, but it is apparently needed in this proof, which I am having trouble to understand.
Firstly, $M$ is a matrix, $v$ is a vector. Define the set $\{Mx | x \cdot v \le 0\}$. So, it is the set of all vectors $Mx$ such that the dot product of each $x$ with $v$ is nonpositive. Note that all vectors involved here have appropiate dimensions so the products all make sense.
I need to show that if this set is unbounded, the following condition is not satisfied.
Condition: There is no $x$ with $x \cdot v\le 0$ such that every element of $Mx$ is nonnegative and at least one is strictly positive.
Firstly, what does it mean that it is bounded, when we're talking a set of vectors? Is it element-wise bounded? Bounded norm? What?
Secondly, how do we show that if the set is unbounded, the condition is not satisfied? The proof I am reading just says:
the set is convex, and so if it is unbounded, there must exist an associative ray which is infinitely long, and therefore the condition is not satisfied.
I understand little of that. Why must there exist an "associative ray"? What actually is it? Why is it infinitely long? In which direction is it infinitely long? And why does its existence prove the condition is not satisfied?