Specifically, I have the following in mind:
- Consider the positive quadrant as the domain (i.e. $(x,y)$ with $x,y\geq 0 $).
- Consider a downward sloping line that divides this domain into two sets (the set that lies below the line and the set that lies above)
Both of these sets are convex.
However, suppose that we introduce a kink in this line. That is, suppose that at some point $x$, the absolute value of the slope increases (so it becomes more downward sloping).
Now the set of points below this line is still convex, but the set of points above the line is no longer convex
- What is the difference between the two sets that causes one to stop being convex and the other remains convex?
I am asking because in the example above we had a straight line. However, I feel like the result would hold even if the boundary was a curve (i.e., if a downward sloping boundary of a convex set suddenly becomes steeper at some point, the set will stop being convex). I have no idea how one would go about proving this, though. (Maybe choose 2 points and find a convex combination not in the set?)
(sorry if kink is not the correct term)