= Quasiconcave function
A \b[quasiconcave function] is a real-valued <function> on a <convex set> whose every <superlevel set> is convex. Equivalently, $-f$ is a <quasiconvex function>, and
$$
f((1-t)x+ty)\geq\min\{f(x),f(y)\},\qquad0\leq t\leq1.
$$
The definition allows empty or whole-domain superlevel sets. Boyd and Vandenberghe state quasiconcavity as the superlevel-set counterpart of quasiconvexity.<Convex Optimization by Stephen Boyd and Lieven Vandenberghe>[§3.4.1, quasiconcave functions]{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>:
$$
f(x)=\frac1{1+x^2},\qquad x\in\mathbb R.
$$
For $0<\alpha\leq1$, its superlevel set is
$$
\{x:f(x)\geq\alpha\}=\left[-\sqrt{\alpha^{-1}-1},\sqrt{\alpha^{-1}-1}\right].
$$
It is all of $\mathbb R$ for $\alpha\leq0$ and empty for $\alpha>1$. These sets are convex intervals. However,
$$
f''(x)=\frac{6x^2-2}{(1+x^2)^3}
$$
is positive when $|x|>1/\sqrt3$, 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.
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