If utility function is convex, what can be said about preference relation?

Question

It is known that if a utility function is concave, then it is quasiconcave, and the preference relation is convex.

What can be said if a utility function is convex?

I've found on the Internet then in this case the function can be both quasiconcave and quasiconvex (how is this possible? linear function?) and is in this case preference relation concave?

I've managed to show that the utility function $u(x_1,x_2)=\operatorname{max}(x_1,x_2)$ is convex. What can be said about the preference relation?

Answer 1

Concave utility functions are quasiconcave while convex utility functions are quasiconvex.

If $U()$ is convex then $-U()$ is concave and represents the 'opposite' preferences, so if you believe the first half of the above statement you can easily show that the second part is true.

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A function is both concave and convex if and only if $$ \alpha f(x) + (1 - \alpha) f(y) = f\left( \alpha x + (1 - \alpha) y \right) $$ for $\alpha \in [0,1]$ and all $x,y$.

E.g.; linear functions have this property.

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Because of the above, linear utility functions (but not only them) will be both quasiconvex and quasiconcave.