# Does quasi-concave utility function imply convex indifference curve?

It is well-known that convex indifference curve (i.e. the function is convex)/ preference would imply quasi-concave utility function. But does quasi-concave utility function imply convex indifference curve?

It seems that this answer give a brief sketch of the proof, but how can I show it in a more formal way?

• Seems like that answer had little effect, because I tried to point out in it that the indifference curve is usually not "convex". Nov 3 '19 at 13:49
• Your question seems fairly straightforward. Have you tried directly applying the definitions of convex preferences and quasi-concave utility functions? Nov 3 '19 at 13:50
• You said that "What you probably mean is that the IC curve implicitly defines a convex function f where f(x)=y." That is what I meant. I am not sure what do you mean by " IC curve is not convex in the usual meaning of the word convex when applied to sets". Nov 3 '19 at 15:48
• Would you mind typing in your quasi-concave utility function to convex preference proof (edit your question, do not add more comments), so we can see why it is not reversible? Nov 3 '19 at 17:10
• On convexity: Most curves are not convex sets. If you pick two points from the curve you can usually find a convex combination of them that is not on the curve. Nov 3 '19 at 17:11

No that is not true. Consider $$u(x, y) = -x^2 - y^2$$ defined on $$\mathbb{R}^2_+$$. Since $$u$$ is concave it is quasiconcave. Observing the graph of the indifference curves, we see that ICs of $$u$$ are not "convex". 