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[[Image:ConvexFunction.svg|thumb|300px|right|Convex function on an [[interval (mathematics)|interval]].]]{{Use American English|date = March 2019}}
{{Short description|Real function with secant line between points above the graph itself}}
[[Image:Epigraph convex.svg|right|thumb|300px|A function (in black) is convex if and only if the region above its [[Graph of a function|graph]] (in green) is a [[convex set]].]]
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[[File:Convex vs. Not-convex.jpg|thumb|right|300px|Convex vs. Not convex]]
In [[mathematics]], a [[real-valued function]] is called '''convex''' if the [[line segment]] between any two distinct points on the [[graph of a function|graph of the function]] lies above the graph between the two points. Equivalently, a function is convex if its [[epigraph (mathematics)|epigraph]] (the set of points on or above the graph of the function) is a [[convex set]]. A twice-[[differentiable function|differentiable]] function of a single variable is convex [[if and only if]] its [[second derivative]] is nonnegative on its entire [[___domain of a function|___domain]].<ref>{{Cite web|url=https://www.stat.cmu.edu/~larry/=stat705/Lecture2.pdf |title=Lecture Notes 2|website=www.stat.cmu.edu|access-date=3 March 2017}}</ref> Well-known examples of convex functions of a single variable include a [[linear function]] <math>f(x) = cx</math> (where <math>c</math> is a [[real number]]), a [[quadratic function]] <math>cx^2</math> (<math>c</math> as a nonnegative real number) and
Convex functions play an important role in many areas of mathematics. They are especially important in the study of [[optimization]] problems where they are distinguished by a number of convenient properties. For instance, a strictly convex function on an [[open set]] has no more than one [[maximum and minimum|minimum]]. Even in infinite-dimensional spaces, under suitable additional hypotheses, convex functions continue to satisfy such properties and as a result, they are the most well-understood functionals in the [[calculus of variations]]. In [[probability theory]], a convex function applied to the [[expected value]] of a [[random variable]] is always bounded above by the expected value of the convex function of the random variable. This result, known as [[Jensen's inequality]], can be used to deduce [[inequality (mathematics)|inequalities]] such as the [[inequality of arithmetic and geometric means|arithmetic–geometric mean inequality]] and [[Hölder's inequality]].
==Definition==
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* A [[differentiable function|differentiable]] function of one variable is convex on an interval if and only if its [[derivative]] is [[monotonically non-decreasing]] on that interval. If a function is differentiable and convex then it is also [[continuously differentiable]].
* A differentiable function of one variable is convex on an interval if and only if its graph lies above all of its [[tangent]]s:<ref name="boyd">{{cite book| title=Convex Optimization| first1=Stephen P.|last1=Boyd |first2=Lieven| last2=Vandenberghe | year = 2004 |publisher=Cambridge University Press| isbn=978-0-521-83378-3| url= https://web.stanford.edu/~boyd/cvxbook/bv_cvxbook.pdf#page=83 |format=pdf | access-date=October 15, 2011}}</ref>{{rp|69}} <math display=block>f(x) \geq f(y) + f'(y) (x-y)</math> for all <math>x</math> and <math>y</math> in the interval.
* A twice differentiable function of one variable is convex on an interval if and only if its [[second derivative]] is non-negative there; this gives a practical test for convexity. Visually, a twice differentiable convex function "curves up", without any bends the other way ([[inflection point]]s). If its second derivative is positive at all points then the function is strictly convex, but the [[
**This property and the above property in terms of "...its derivative is monotonically non-decreasing..." are not equal since if <math>f''</math> is non-negative on an interval <math>X</math> then <math>f'</math> is monotonically non-decreasing on <math>X</math> while its converse is not true, for example, <math>f'</math> is monotonically non-decreasing on <math>X</math> while its derivative <math>f''</math> is not defined at some points on <math>X</math>.
* If <math>f</math> is a convex function of one real variable, and <math>f(0)\le 0</math>, then <math>f</math> is [[Superadditivity|superadditive]] on the [[positive reals]], that is <math>f(a + b) \geq f(a) + f(b)</math> for positive real numbers <math>a</math> and <math>b</math>.
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* A function <math>f</math> is midpoint convex on an interval <math>C</math> if for all <math>x_1, x_2 \in C</math> <math display=block>f\!\left(\frac{x_1 + x_2}{2}\right) \leq \frac{f(x_1) + f(x_2)}{2}.</math> This condition is only slightly weaker than convexity. For example, a real-valued [[Lebesgue measurable function]] that is midpoint-convex is convex: this is a theorem of [[Wacław Sierpiński|Sierpiński]].<ref>{{cite book|last=Donoghue|first=William F.| title= Distributions and Fourier Transforms|year=1969|publisher=Academic Press | isbn=9780122206504 |url= https://books.google.com/books?id=P30Y7daiGvQC&pg=PA12|access-date=August 29, 2012|page=12}}</ref> In particular, a continuous function that is midpoint convex will be convex.
=== Functions of several variables ===
* A function <math>f : X \to [-\infty, \infty]</math> valued in the [[extended real
* A differentiable function <math>f</math> defined on a convex ___domain is convex if and only if <math>f(x) \geq f(y) + \nabla f(y)^T \cdot (x-y)</math> holds for all <math>x, y</math> in the ___domain.
* A twice differentiable function of several variables is convex on a convex set if and only if its [[Hessian matrix]] of second [[partial derivative]]s is [[Positive-definite matrix|positive semidefinite]] on the interior of the convex set.
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