Quasiconvex function

Remarks

Equivalently, for each $a \in \R$ the sublevel set $\set{x \in S : f(x) \le a}$ is convex; compare level sets. The function $f$ is quasiconcave if $-f$ is quasiconvex. A convex function is quasiconvex, because $t f(x) + (1 - t) f(y) \le \max\set{f(x), f(y)}$.

Example: $f(x) = \sqrt{\abs{x}}$ on $\R$. Its sublevel sets are $\emptyset$ for $a < 0$ and $[-a^2, a^2]$ for $a \ge 0$, all convex. It is not convex, since $f(1/2) = \sqrt{1/2} > 1/2 = \frac{1}{2} f(0) + \frac{1}{2} f(1)$.

Non-example: $f(x) = -x^2$ on $\R$. With $x = -1$, $y = 1$ and $t = 1/2$, one gets $f(0) = 0 > -1 = \max\set{f(-1), f(1)}$.