I need an example of a
Convex, non-monotonic preference
Non-convex, monotonic preference
I figured that an example of non-convex, monotonic utility preference could be $U(x,y)=x^2+y^2$. For convex, non-monotonic preference, I cannot think of a standard example. As far as my understanding goes, lexicographic preferences are convex, but I don't think these are non monotonic. Please help me figure out the examples so that I can conceptually understand monotonicity and convexity.