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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 of a single variable is convex [[if and only if]] its second derivative is nonnegative on its entire ___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 the [[quadratic function]] <math>x^2</math> and the [[exponential function]] <math>e^x</math>. In simple terms, a convex function refers to a function whose graph is shaped like a cup <math>\cup</math>, while a [[concave function]]'s graph is shaped like a cap <math>\cap</math>.
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 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 inequalities such as the [[inequality of arithmetic and geometric means|arithmetic–geometric mean inequality]] and [[Hölder's inequality]].
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