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- A concave function can be quasiconvex. For example, x ↦ log (x) {displaystyle xmapsto log (x)} is both concave and quasiconvex. Any monotonic function is both quasiconvex and quasiconcave. More generally, a function which decreases up to a point and increases from that point on is quasiconvex (compare unimodality).
en.wikipedia.org/wiki/Quasiconvex_function
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For example $f(\mathbf{X}) = \text{rank}(\mathbf{X})$ is a quasi-concave on $\mathbb{S}^n_{+}$. Ceiling function $f(x)= \lceil x \rceil$ is a quasi-concave function (also, it is quasi-convex which is called quasi-linear).
For example, f(x) transformation, but g(f(x)) = = −x2 is concave, and g(x) 2 = ex is a monotonic e−x2. 2 is not concave. This is problematic when we want to analyze things like utility which we consider to be ordinal concepts. A weaker condition to describe a function is quasiconvexity (or quasiconcav-ity).
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A function is concave (convex) if the graph of the function is always above (below) any chord (line segment between two points in the graph). Remark 4. f concave ⇔−f convex. Example 5. Let S = [0,∞) and consider f(x) = √ x and g(x) = −f(x) = − √ x f is a concave function and g is a convex function. 1.2 Selected properties of ...
convex function is quasiconvex. Every c. Prove that the following statements are equivalent. The function f : C R is quasiconvex. →. (b) For all α R, the sublevel set x C : f(x) α is convex. ∈ { ∈ ⩽ } 7–2. (c) For all α R, the strict sublevel set x C : f(x) < α is convex. ∈ { ∈ }
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In mathematics, a quasiconvex function is a real-valued function defined on an interval or on a convex subset of a real vector space such that the inverse image of any set of the form (,) is a convex set. For a function of a single variable, along any stretch of the curve the highest point is one of the endpoints.
Definition: A function is strictly quasiconvex if all of its lower contour sets are strictly convex sets and none of its level sets have any width (i.e., no interior). The first condition rules out straight-line level sets while the second rules out flat spots.
Note that f (x) = g(u(x; y)) where u(x; y) = xy and g(z) = z + z2 + z3. Since g0 > 0, f is quasi-concave if and only if u is quasi-concave. But u(x; y) = ev(x;y) where v(x; y) = ln x + ln y. The function v is easily seen to be concave. So then. (x) = g(u(x; y)) = g(ev(x;y))
