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A function f is concave over a convex set if and only if the function −f is a convex function over the set. The sum of two concave functions is itself concave and so is the pointwise minimum of two concave functions, i.e. the set of concave functions on a given domain form a semifield.
In simple terms, a convex function 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 . A twice- differentiable function of a single variable is convex if and only if its second derivative is nonnegative on its entire domain . [ 1 ]
A log-concave function is also quasi-concave. This follows from the fact that the logarithm is monotone implying that the superlevel sets of this function are convex. [1] Every concave function that is nonnegative on its domain is log-concave. However, the reverse does not necessarily hold.
A convex function is a real-valued function defined on an interval with the property that its epigraph (the set of points on or above the graph of the function) is a convex set. Convex minimization is a subfield of optimization that studies the problem of minimizing convex functions over convex sets.
For example, the problem of maximizing a concave function can be re-formulated equivalently as the problem of minimizing the convex function . The problem of maximizing a concave function over a convex set is commonly called a convex optimization problem. [8]
Jensen's inequality generalizes the statement that a secant line of a convex function lies above its graph. Visualizing convexity and Jensen's inequality. In mathematics, Jensen's inequality, named after the Danish mathematician Johan Jensen, relates the value of a convex function of an integral to the integral of the convex function.
A logarithmically convex function f is a convex function since it is the composite of the increasing convex function and the function , which is by definition convex.However, being logarithmically convex is a strictly stronger property than being convex.
Any function that is concave and continuous, and defined on a set that is convex and compact, attains its minimum at some extreme point of that set. Since a linear function is simultaneously convex and concave, it satisfies both principles, i.e., it attains both its maximum and its minimum at extreme points.