Concave function

In mathematics, a concave function is one for which the function value at any convex combination of elements in the domain is greater than or equal to that convex combination of those domain elements. 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 convex functions. A concave function is also synonymously called concave downwards, concave down, convex upwards, convex cap, or upper convex.

x and f((1−α)x+αy)≥(1−α)f(x)+αf(y)

A function is called strictly concave if

(1-\\alpha ) f(x)+\\alpha f(y)"}}'>f((1−α)x+αy)>(1−α)f(x)+αf(y)

for any f:ℝ→ℝ, this second definition merely states that for every y, the point (x,f(x)) and S(a)={x:f(x)≥a} are convex sets.

A cubic function is concave (left half) when its first derivative (red) is monotonically decreasing i.e. its second derivative (orange) is negative, and convex (right half) when its first derivative is monotonically increasing i.e. its second derivative is positive

1. A differentiable function f is (strictly) concave on an interval if and only if its derivative function f is concave if and only if f is strictly concave, but the converse is not true, as shown by 4</sup>"}},"i":0}}]}'/>f(x) = −x.
2. If C is concave if and only if it is midpoint concave, that is, for any C f(0) ≥ 0, then f is concave and f(tx)=f(tx+(1−t)⋅0)≥tf(x)+(1−t)f(0)≥tf(x).
3. For f is concave over a convex set if and only if the function g(x)=x are concave on their domains, as their second derivatives f(x)=log⁡x is concave on its domain f(x)=ax+b is both concave and convex, but neither strictly-concave nor strictly-convex.
4. The sine function is concave on the interval |B| is the determinant of a nonnegative-definite matrix B, is concave.

Applications

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• 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.
• In thermodynamics and information theory, entropy is a concave function. In the case of thermodynamic entropy, without phase transition, entropy as a function of extensive variables is strictly concave. If the system can undergo phase transition, and if it is allowed to split into two subsystems of different phase (phase separation, e.g. boiling), the entropy-maximal parameters of the subsystems will result in a combined entropy precisely on the straight line between the two phases. This means that the "effective entropy" of a system with phase transition is the convex envelope of entropy without phase separation; therefore, the entropy of a system including phase separation will be non-strictly concave.




See also

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

References

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Varian, Hal R. (1992). Microeconomic analysis (3rd ed.). New York: Norton. p. 489. ISBN 0-393-95735-7. OCLC 24847759.
↑ Hass, Joel (13 March 2017). Thomas' calculus. Heil, Christopher, 1960-, Weir, Maurice D.,, Thomas, George B. Jr. (George Brinton), 1914-2006. (Fourteenth ed.). [United States]. p. 203. ISBN 978-0-13-443898-6. OCLC 965446428.{{cite book}}: CS1 maint: location missing publisher (link)
↑ Callen, Herbert B. (1985). "8.1: Intrinsic Stability of Thermodynamic Systems". Thermodynamics and an Introduction to Thermostatistics (2nd ed.). New York: Wiley. pp. 203–206. ISBN 978-0-471-86256-7.{{cite book}}: CS1 maint: date and year (link)