Convex Function

Convex Function

In mathematics, a real-valued function defined on an interval is called convex (or convex downward or concave upward) if the graph of the function lies below the line segment joining any two points of the graph; or more generally, any two points in a vector space. Equivalently, a function is convex if its epigraph (the set of points on or above the graph of the function) is a convex set. Well-known examples of convex functions are the quadratic function and the exponential function for any real number x.

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 less or equal to the expected value of the convex function of the random variable. This result, known as Jensen's inequality underlies many important inequalities (including, for instance, the arithmetic-geometric mean inequality and Hölder's inequality).

Exponential growth is a special case of convexity. Exponential growth narrowly means "increasing at a rate proportional to the current value", while convex growth generally means "increasing at an increasing rate (but not necessarily proportionally to current value)".