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Concept
Convex function
Summary
In mathematics, a real-valued function is called convex if the line segment between any two distinct points on the graph of the function lies above the graph between the two points. Equivalently, a function is convex if its 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. Well-known examples of convex functions of a single variable include a linear function (where is a real number), a quadratic function ( as a nonnegative real number) and a exponential function ( as a nonnegative real number). In simple terms, a convex function refers to a function whose graph is shaped like a cup (or a straight line like a linear function), while a concave function's graph is shaped like a cap .
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 arithmetic–geometric mean inequality and Hölder's inequality.
Let be a convex subset of a real vector space and let be a function.
Then is called if and only if any of the following equivalent conditions hold:
For all and all :
The right hand side represents the straight line between and in the graph of as a function of increasing from to or decreasing from to sweeps this line.
Official source
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