Let X be a convex subset of linear topological space and let binary relation >= be a complete preordering.
prove: If preference relation is strictly convex and continuous, then it is convex.
Since the definition of weakly convex preference is equivalent to weak upper contour sets and strict upper contour set being convex. The relationship between convex and weakly convex preference can be easily established. But there is no such relationships exist with convex preference. I have no idea how to prove the above mentioned proposition. thanks!