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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.

Unimodality, in the single-maximum convention, means that a real-valued function rises toward one mode or plateau and then falls away. More precisely, for a function $f$ on an interval, there is a point $m$ such that $f$ is nondecreasing to the left of $m$ and nonincreasing to its right. These inequalities are weak: a flat top is allowed, and more than one point may achieve the maximum.

For an original example,is zero throughout $[-1,1]$ and decreases as one moves outward beyond that plateau. It is unimodal despite lacking a unique maximizing point. In contrast, two separated peaks with a genuine dip between them violate the rise-then-fall requirement.

A useful characterization is that a unimodal function in this convention has interval-shaped superlevel sets $\{x:f(x)\ge c\}$. For a function attaining its maximum, the converse holds too: if all those sets are intervals, a decline followed by a later rise on the same side of a maximizer would create a disconnected superlevel set. In one dimension this is the quasiconcave function viewpoint. Boyd and Vandenberghe define quasiconcavity through convex superlevel sets.^[ref]

The convention must be stated. In minimization, unimodal is also used for a function that first decreases and then increases; that corresponds to a quasiconvex function and interval-shaped sublevel sets. A single minimum and a single maximum are opposite shape conditions.

Unimodality also differs from concavity. The function $g(x)=e^{-x^2}$ has derivative $-2x{e}^{-x^2}$, so it rises up to zero and falls afterward. However,so it is not a concave function on the whole real line. Unimodality constrains the ordering of levels rather than the sign of curvature everywhere.

For a probability density, this shape defines one common meaning of a unimodal distribution. Adding symmetry around zero strengthens the conclusion: the centred interval maximizes mass for an even unimodal density lemma shows why a fixed-width observation window collects most probability when centered on the mode.