In mathematics, a concave function is one for which the value at any convex combination of elements in the domain is greater than or equal to the convex combination of the values at the endpoints. Equivalently, a concave function is any function for which the hypograph is convex. The class of concave functions is in a sense the opposite of the class of a convex functions. A concave function is also synonymously called concave downwards, concave down, convex upwards, convex cap, or upper convex.

Definition

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]

Properties

Examples

• 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]

Applications

• 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]

See also

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

References

Further References

• 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.