Doesnt convexity prevent thick indifference curves aswell? 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.
My approach would be to define thick indifference curves in terms of a stronger form of local non-satiation:


  Definition (thick indifference curves) Preferences are said to have thick indifference curves if there exists at least one bundle $A\in\mathbb{R}^l$ and an open ball
  $\mathscr{B}(A)$ around $A$ such that $A'\sim A$ for every
  $A'\in\mathscr{B}(A)$.


Let's set up a definition for strict convexity to be sure we are on the same page:


  Definition (strict convexity) Preferences obey strict convexity if
  
  $$A\sim B\implies \lambda A+(1-\lambda) B\succ A\sim B$$
  
  for $A,B\in\mathbb{R}^l$ and any $\lambda\in(0,1)$.


Now we can state the desired result:


  Proposition If $\succsim$ is strictly convex then $\succsim$ does not have thick indifference curves.


Proof Suppose that $\succsim$ has thick indifference curves. Then there exists an $A$ such that $A'\sim A$ for all $A'\in\mathscr{B}(A)$. Thus, fix two bundles $A$ and $B$ such that $A\sim B$ and $B\in\mathscr{B}(A)$. We know that 
$$\lambda A+(1-\lambda) B \in \mathscr{B}(A).$$
Thus, if the indifference curve is thick around point$A$ we must have $\lambda A+(1-\lambda) B\sim A\sim B$. But this contradicts the strict convexity hypothesis. QED



That should do the job for strict convexity. As frage_man already pointed out, the proposition fails to hold generally under weak convexity. Indifference everywhere is a very elegant example why.