In Mas-Colell, Whinston, and Green's Microeconomics they define the indirect utility function, $v(p,w)$ as
$$ v(p,w) := u(x^*) $$
Where $x^* \in x(p,w)$ solves the utility maximization problem.
They state a property of $v(p,w)$ is quasiconvexity, i.e. the set
$$ \{(p,w): v(p,w) \leq \bar{v} \} $$
is convex for any $\bar{v}$.
Just the page before they said that convexity of preferences implies that $u(\bullet)$ is quasiconcave, so my question is why when we look at the max of $u$, it's quasiconcave property inflects (can't think of a better word) to quasiconvexity?