Preference: Convexity and Monotonicity 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.
$u_1(x, y) = -(x-1)^2 - (y-1)^2$ 

$u_2(x, y) = -|x-1| - |y-1|$

are examples of utility functions that are concave, and therefore represent the preferences that are convex. Also, these preferences are not monotonic.