In mathematics, convex function is a real-valued function f defined on an interval (or on any convex subset C of some vector space), if for any two points x and y in its domain C and any t in [0,1], we have Euclid, a famous Greek mathematician known as the father of geometry, is shown here in detail from The School of Athens by Raphael. ... Partial plot of a function f. ... In mathematics, interval is a concept relating to the sequence and set-membership of one or more numbers. ... In mathematics, an object is convex if for any pair of points within the object, any point on the straight line segment that joins them is also within the object. ... Vector spaces (or linear spaces) are spaces whose elements, known as vectors, can be scaled and added; all linear combinations can be formed. ... In mathematics, the domain of a function is the set of all input values to the function. ...

Convex function on an interval.

In other words, a function is convex if and only if its epigraph (the set of points lying on or above the graph) is a convex set. A function is also said to be strictly convex if Replacement for original convex function graph File links The following pages link to this file: Convex function Categories: GFDL images ... Replacement for original convex function graph File links The following pages link to this file: Convex function Categories: GFDL images ... It has been suggested that this article or section be merged with Logical biconditional. ... In literature, an epigraph is a quotation that is placed at the start of a work or section that expresses in some succinct way an aspect or theme of what is to follow. ... Look up Convex set in Wiktionary, the free dictionary. ...

for any t in (0,1).

The opposite of a convex function is a concave function. The additive inverse, or opposite, of a number n is the number which, when added to n, yields zero. ... In calculus, a differentiable function f is convex on an interval if its derivative function f ′ is increasing on that interval: a convex function has an increasing slope. ...

Properties of convex functions

A convex function f defined on some convex open interval C is continuous on C and differentiable at all but at most countably many points. If C is closed, then f may fail to be continuous at the endpoints of C. In elementary algebra, an interval is a set that contains every real number between two indicated numbers, and possibly the two numbers themselves. ... In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. ... In mathematics, the derivative is defined as the instantaneous rate of change of a function. ... In mathematics the term countable set is used to describe the size of a set, e. ...

A continuous function on an interval C is convex if and only if

for all x and y in C.

A differentiable function of one variable is convex on an interval if and only if its derivative is monotonically non-decreasing on that interval. In mathematics, the derivative of a function is one of the two central concepts of calculus. ...

A continuously differentiable function of one variable is convex on an interval if and only if the function lies above all of its tangents: f(y) ≥ f(x) + f'(x) (y − x) for all x and y in the interval. In mathematics, a smooth function is one that is infinitely differentiable, i. ... This article is about the mathematical concept of tangent. For other meanings, see tangent (disambiguation). ...

A twice differentiable function of one variable is convex on an interval if and only if its second derivative is non-negative there; this gives a practical test for convexity. If its second derivative is positive then it is strictly convex, but the opposite is not true, as shown by f(x) = x⁴.

More generally, a continuous, twice differentiable function of several variables is convex on a convex set if and only if its Hessian matrix is positive semidefinite on the interior of the convex set. In mathematics, the Hessian matrix is the square matrix of second partial derivatives of a scalar-valued function. ... In linear algebra, a positive-definite matrix is a Hermitian matrix which in many ways is analogous to a positive real number. ...

If two functions f and g are convex, then so is any weighted combination a f + b g with non-negative coefficients a and b. Likewise, if f and g are convex, then the function max{f,g} is convex.

Any local minimum of a convex function is also a global minimum. A strictly convex function will have at most one global minimum. A graph illustrating local min/max and global min/max points In mathematics, a point x* is a local maximum of a function f if there exists some ε > 0 such that f(x*) ≥ f(x) for all x with |x-x*| < ε. ... A graph illustrating local min/max and global min/max points In mathematics, a point x* is a local maximum of a function f if there exists some ε > 0 such that f(x*) ≥ f(x) for all x with |x-x*| < ε. ...

For a convex function f, the level sets {x | f(x) < a} and {x | f(x) ≤ a} with a ∈ R are convex sets. In mathematics, a level set of a real-valued function f of n variables is a set of the form { (x1,...,xn) | f(x1,...,xn) = c } where c is a constant. ...

Convex functions respect Jensen's inequality. In mathematics, Jensens 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. ...

Examples

• The second derivative of x² is 2; it follows that x² is a convex function of x.
• The absolute value function |x| is convex, even though it does not have a derivative at x = 0.
• The function f with domain [0,1] defined by f(0)=f(1)=1, f(x)=0 for 0<x<1 is convex; it is continuous on the open interval (0,1), but not continuous at 0 and 1.
• The function x³ has second derivative 6x; thus it is convex for x ≥ 0 and concave for x ≤ 0.
• Every linear transformation is convex but not strictly convex, since if f is linear, then f(a + b) = f(a) + f(b). This implies that the identity map (i.e., f(x) = x) is convex but not strictly convex. The fact holds if we replace "convex" by "concave".
• An affine function is simultaneously convex and concave.

In mathematics, the absolute value (or modulus1) of a real number is its numerical value without regard to its sign. ... In calculus, a differentiable function f is convex on an interval if its derivative function f ′ is increasing on that interval: a convex function has an increasing slope. ... In mathematics, a linear transformation (also called linear map or linear operator) is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. ... An identity function f is a function which doesnt have any effect: it always returns the same value that was used as its argument. ... The word linear comes from the Latin word linearis, which means created by lines. ...

See also

• Logarithmically convex function
• Subderivative of a convex function