Concave function

In mathematics, a concave function is the negative of a convex function. A concave function is also synonymously called concave downwards, concave down, convex upwards, convex cap, or upper convex.

Definition

A real-valued function $f$ on an interval (or, more generally, a convex set in vector space) is said to be ''concave'' if, for any $x$ and $y$ in the interval and for any \alpha \in ,1/math>,

:$f((1-\alpha )x+\alpha y)\geq (1-\alpha ) f(x)+\alpha f(y)$

A function is called ''strictly concave'' if

:$f((1-\alpha )x + \alpha y) > (1-\alpha) f(x) + \alpha f(y)\,$

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

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

A function $f$ is quasiconcave if the upper contour sets of the function $S(a)=\$ are convex sets.

Properties

Functions of a single variable

# A differentiable function is (strictly) concave on an interval if and only if its derivative function is (strictly) monotonically decreasing on that interval, that is, a concave function has a non-increasing (decreasing) slope.
# Points where concavity changes (between concave and convex) are inflection points.
# If is twice-differentiable, then is concave if and only if is non-positive (or, informally, if the "acceleration" is non-positive). If its second derivative is negative then it is strictly concave, but the converse is not true, as shown by .
# If is concave and differentiable, then it is bounded above by its first-order Taylor approximation: f(y) \leq f(x) + f'(x) -x/math>
# A Lebesgue measurable function on an interval is concave if and only if it is midpoint concave, that is, for any and in $f\left( \frac2 \right) \ge \frac2$
# If a function is concave, and , then is subadditive on $[0,\infty)$. Proof:
#* Since is concave and , letting we have $f(tx) = f(tx+(1-t)\cdot 0) \ge t f(x)+(1-t)f(0) \ge t f(x) .$
#* For $a,b\in[0,\infty)$: $f(a) + f(b) = f \left((a+b) \frac \right) + f \left((a+b) \frac \right) \ge \frac f(a+b) + \frac f(a+b) = f(a+b)$

Functions of ''n'' variables

# A function is concave over a convex set if and only if the function is a convex function over the set.
# The sum of two concave functions is itself concave and so is the pointwise minimum of two concave functions, i.e. the set of concave functions on a given domain form a semifield.
# Near a local maximum in the interior of the domain of a function, the function must be concave; as a partial converse, if the derivative of a strictly concave function is zero at some point, then that point is a local maximum.
# Any local maximum of a concave function is also a global maximum. A ''strictly'' concave function will have at most one global maximum.

Examples

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

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.

See also

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

References

Further References

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{{Convex analysis and variational analysis

Convex analysis
Types of functions