Quasiconcave function

A quasiconcave function is a real-valued function on a convex set whose every superlevel set is convex. Equivalently, $-f$ is a quasiconvex function, andThe definition allows empty or whole-domain superlevel sets. Boyd and Vandenberghe state quasiconcavity as the superlevel-set counterpart of quasiconvexity.^[ref]

For the equivalence, choose the level $\alpha =\min \{f(x),f(y)\}$. Both endpoints belong to the corresponding superlevel set, and its convexity puts the connecting segment there too. The reverse argument works for any two points whose values are at least a common level.

An original example separates quasiconcavity from concavity:For $0<\alpha \le 1$, its superlevel set isIt is all of $\mathbb{R}$ for $\alpha \le 0$ and empty for $\alpha >1$. These sets are convex intervals. However,is positive when $|x|>1/\sqrt{3}$, so $f$ is not a concave function on the real line.

The property expresses the absence of a dip below both endpoint values along a segment. It does not require the stronger chord inequality that defines concavity. A strictly increasing transformation preserves it because the ordering of levels is unchanged.

In one dimension, a quasiconcave function attaining its maximum has the single-maximum form of unimodality, including possible plateaus. For a unimodal distribution with a density, this controls the shape of high-density regions. Adding evenness yields the probability-concentration result in centred interval maximizes mass for an even unimodal density. Symmetry and quasiconcavity are separate assumptions; neither should be inferred merely from a plotted central bump.