It is standard in many micro textbooks when analyzing the relationship between preference axioms and the shape of the utility function (and consequently the shape of indifference curves), to attributed the "non-thickness" of indifference curves to "local non-satiation". While it is easily seen (and proven) that LNS preferences don't admit thick IC, my question is the following:
Thick IC violate as well strict convexity (notice that weak convexity appears to "survive"), so, isn't convexity alone another property that prevents thickness? Put it differently, can we find non monotonic, convex preferences that don't admit thick IC?
If any proof or reference to relevant literature would be very useful.