Convex function: Difference between revisions

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In simple terms, a convex function graph is shaped like a cup <math>\cup</math> (or a straight line like a linear function), while a [[concave function]]'s graph is shaped like a cap <math>\cap</math>.

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 positivenonnegative real number) and an [[exponential function]] <math>ce^x</math> (<math>c</math> as a positivenonnegative real number).

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&ndash;geometric mean inequality]] and [[Hölder's inequality]].