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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.

Quasiconvex optimization minimizes a quasiconvex function over a convex set of feasible points. Its defining advantage is that every sublevel set is convex: for each fixed threshold $t$, asking whether a feasible $x$ satisfies $f(x)\le t$ becomes a convex feasibility problem. The objective function itself need not be a convex function.^[ref]

For an original one-dimensional example, minimizeIts derivative $f^{\prime }(x)=1/(x+2{)}^2$ is positive, so its sublevel sets on this interval are intervals and the function is a quasiconvex function. Yet $f^{\prime \prime }(x)=-2/(x+2{)}^3<0$, so this nonconstant function is strictly concave rather than convex. For fixed $t$, the denominator is positive, and the threshold condition becomes the linear inequalityAt $t=0.4$, no allowed $x$ satisfies it; at $t=0.6$, the allowed interval $0\le x\le 0.5$ does. The true minimum is $f(0)=0.5$.

Interval bisection can search thresholds without directly minimizing the original ratio. Maintain an infeasible lower threshold $L$ and a feasible upper threshold $U$, with a stored feasible witness for $U$. Test $t=(L+U)/2$: on feasibility replace $U$ and its witness; on infeasibility replace $L$. With exact decisions, after $k$ tests the bracket width isStarting with $L=0$, $U=1$ and witness $x=2$, the example's first test finds $x=0$ feasible at $t=0.5$. Later tests below0.5tighten the lower bound.^[ref]

The threshold search is only as reliable as its feasibility problem solver and tolerances. Convexity of the sublevel sets does not supply an inexpensive representation of every such set, and the interval-width guarantee assumes justified feasibility decisions. Returning a feasible witness also matters: an objective bound alone is not an implementable solution.

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.