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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!

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    $\begingroup$ You may want to double-check the statement you're asked to prove. Seems continuity is redundant here. $\endgroup$ – Herr K. Mar 19 at 18:27
  • $\begingroup$ But continuity is a regularity condition on contour sets which is important in this context. $\endgroup$ – deepanshu Mar 20 at 5:54
  • $\begingroup$ Can you please add the definitions you are using? Terminology varies a bit. $\endgroup$ – Michael Greinecker Mar 31 at 13:47
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    $\begingroup$ @HerrK. For the definitions used by Debreu in his "Theory of Value," the condition is not redundant. $\endgroup$ – Michael Greinecker Mar 31 at 13:53

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