Concave function

A real-valued function ${\displaystyle f}$ on an interval (or, more generally, a convex set in vector space) is said to be concave if, for any ${\displaystyle x}$ and ${\displaystyle y}$ in the interval and for any ${\displaystyle \alpha \in [0,1]}$,[1]

${\displaystyle f((1-\alpha )x+\alpha y)\geq (1-\alpha )f(x)+\alpha f(y)}$

A function is called strictly concave if

${\displaystyle f((1-\alpha )x+\alpha y)>(1-\alpha )f(x)+\alpha f(y)\,}$

for any ${\displaystyle \alpha \in (0,1)}$ and ${\displaystyle x\neq y}$.

For a function ${\displaystyle f:\mathbb {R} \to \mathbb {R} }$, this second definition merely states that for every ${\displaystyle z}$ strictly between ${\displaystyle x}$ and ${\displaystyle y}$, the point ${\displaystyle (z,f(z))}$ on the graph of ${\displaystyle f}$ is above the straight line joining the points ${\displaystyle (x,f(x))}$ and ${\displaystyle (y,f(y))}$.

A function ${\displaystyle f}$ is quasiconcave if the upper contour sets of the function ${\displaystyle S(a)=\{x:f(x)\geq a\}}$ are convex sets.[2]

• The functions ${\displaystyle f(x)=-x^{2}}$ and ${\displaystyle g(x)={\sqrt {x}}}$ are concave on their domains, as their second derivatives ${\displaystyle f''(x)=-2}$ and ${\textstyle g''(x)=-{\frac {1}{4x^{3/2}}}}$ are always negative.
• The logarithm function ${\displaystyle f(x)=\log {x}}$ is concave on its domain ${\displaystyle (0,\infty )}$, as its derivative ${\displaystyle {\frac {1}{x}}}$ is a strictly decreasing function.
• Any affine function ${\displaystyle f(x)=ax+b}$ is both concave and convex, but neither strictly-concave nor strictly-convex.
• The sine function is concave on the interval ${\displaystyle [0,\pi ]}$.
• The function ${\displaystyle f(B)=\log |B|}$, where ${\displaystyle |B|}$ is the determinant of a nonnegative-definite matrix B, is concave.[6]

• Rays bending in the computation of radiowave attenuation in the atmosphere involve concave functions.
• In expected utility theory for choice under uncertainty, cardinal utility functions of risk averse decision makers are concave.
• In microeconomic theory, production functions are usually assumed to be concave over some or all of their domains, resulting in diminishing returns to input factors.[7]

• Concave polygon
• Jensen's inequality
• Logarithmically concave function
• Quasiconcave function
• Concavification

• Crouzeix, J.-P. (2008). "Quasi-concavity". In Durlauf, Steven N.; Blume, Lawrence E (eds.). The New Palgrave Dictionary of Economics (Second ed.). Palgrave Macmillan. pp. 815–816. doi:10.1057/9780230226203.1375. ISBN 978-0-333-78676-5.
• Rao, Singiresu S. (2009). Engineering Optimization: Theory and Practice. John Wiley and Sons. p. 779. ISBN 978-0-470-18352-6.