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Negative of a convex function

In [ mathematics ](/wiki/Mathematics "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 ](/wiki/Hypograph_\(mathematics\) "Hypograph \(mathematics\)") is
convex. The class of concave functions is in a sense the opposite of the class
of a [ convex functions ](/wiki/Convex_function "Convex function") . A concave
function is also [ synonymously ](/wiki/Synonym "Synonym") called **concave
downwards** , **concave down** , **convex upwards** , **convex cap** , or
**upper convex** .

##  Definition  [  [ edit
](/w/index.php?title=Concave_function&action=edit&section=1 "Edit section:
Definition") ]

A real-valued [ function ](/wiki/Function_\(mathematics\) "Function
\(mathematics\)") f  {\displaystyle f}  ![{\\displaystyle
f}](https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61)
on an [ interval ](/wiki/Interval_\(mathematics\) "Interval \(mathematics\)")
(or, more generally, a [ convex set ](/wiki/Convex_set "Convex set") in [
vector space ](/wiki/Vector_space "Vector space") ) is said to be _concave_
if, for any  x  {\displaystyle x}  ![{\\displaystyle
x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4)
and  y  {\displaystyle y}  ![{\\displaystyle
y}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d)
in the interval and for any  α  ∈  [  0  ,  1  ]  {\displaystyle \alpha \in
[0,1]}  ![{\\displaystyle \\alpha \\in
\[0,1\]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/daf3c62599ea71319c85f715c9e590d2bab2d036)
,  [1]

     f  (  (  1  −  α  )  x  \+  α  y  )  ≥  (  1  −  α  )  f  (  x  )  \+  α  f  (  y  )  {\displaystyle f((1-\alpha )x+\alpha y)\geq (1-\alpha )f(x)+\alpha f(y)}  ![{\\displaystyle f\(\(1-\\alpha \)x+\\alpha y\)\\geq \(1-\\alpha \)f\(x\)+\\alpha f\(y\)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6f90ff17a492a022ca98d045d9975893e524f66d)

A function is called _strictly concave_ if

     f  (  (  1  −  α  )  x  \+  α  y  )  > (  1  −  α  )  f  (  x  )  \+  α  f  (  y  )  {\displaystyle f((1-\alpha )x+\alpha y)>(1-\alpha )f(x)+\alpha f(y)\,}  ![{\\displaystyle f\(\(1-\\alpha \)x+\\alpha y\)>\(1-\\alpha \)f\(x\)+\\alpha f\(y\)\\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2e752449bb5a2241ddcd11bf7b7f7cd7e0a5b087)

for any  α  ∈  (  0  ,  1  )  {\displaystyle \alpha \in (0,1)}
![{\\displaystyle \\alpha \\in
\(0,1\)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5df576f7940384416d1553ab063704d37bf99420)
and  x  ≠  y  {\displaystyle x\neq y}  ![{\\displaystyle x\\neq
y}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f51b711ca7f932963cdb268b0817dc72d6258733)
.

For a function  f  :  R  →  R  {\displaystyle f:\mathbb {R} \to \mathbb {R} }
![{\\displaystyle f:\\mathbb {R} \\to \\mathbb {R}
}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1e3a10a3ad05781f5cf9c2d875a02227e21a8448)
, this second definition merely states that for every  z  {\displaystyle z}
![{\\displaystyle
z}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98)
strictly between  x  {\displaystyle x}  ![{\\displaystyle
x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4)
and  y  {\displaystyle y}  ![{\\displaystyle
y}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d)
, the point  (  z  ,  f  (  z  )  )  {\displaystyle (z,f(z))}
![{\\displaystyle
\(z,f\(z\)\)}](https://wikimedia.org/api/rest_v1/media/math/render/s