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- Yes! Both linear and affine functions are both convex and concave.
faculty.fiu.edu/~boydj/mathecon/math21.pdf
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A function $f(x)$ is said to be quasi-concave if its domain and all its $\alpha$-superlevel sets defined as $$\mathcal{S}_\alpha \triangleq\{x|x\in dom f, f(x)\geq\alpha\}$$ are convex for every $\alpha$. Another representation of quasi-concave functions: $f(x)$ is called quasi- concave if and only if
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 ...
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.
KC Border Quasiconvex Functions I 7–3 7.2.4 Example Note that strict quasiconvexity implies Condition (E). The con-verse does not hold. In fact, Condition (E) alone does not guarantee even quasi-convexity. For instance, the function f: Rm → R defined by f(x) = 1 x = 0, 0 x ̸= 0 . is not quasiconvex, since the sublevel set {f < 1} = Rm \{0 ...
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Yes! Both linear and affine functions are both convex and concave. Example 21.1.2: Convex Functions. Any linear function f(x) = p·x is both concave and convex, as is the generic affine function f(x) = a+p·x.
• The Conjugate Function • Quasiconvex Functions • Log-concave and Log-convex Functions • Convexity with respect to Generalized Inequalities
