
In
mathematics, a
real-valued function
In mathematics, a real-valued function is a function whose values are real numbers. In other words, it is a function that assigns a real number to each member of its domain.
Real-valued functions of a real variable (commonly called ''real ...
is called convex if the
line segment
In geometry, a line segment is a part of a straight line that is bounded by two distinct end points, and contains every point on the line that is between its endpoints. The length of a line segment is given by the Euclidean distance between ...
between any two 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
In geometry, a subset of a Euclidean space, or more generally an affine space over the reals, is convex if, given any two points in the subset, the subset contains the whole line segment that joins them. Equivalently, a convex set or a convex ...
. A twice-differentiable function of a single variable is convex
if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false.
The connective is bi ...
its second derivative is nonnegative on its entire domain. Well-known examples of convex functions of a single variable include the
quadratic function
In mathematics, a quadratic polynomial is a polynomial of degree two in one or more variables. A quadratic function is the polynomial function defined by a quadratic polynomial. Before 20th century, the distinction was unclear between a polynomi ...
and the
exponential function
The exponential function is a mathematical function denoted by f(x)=\exp(x) or e^x (where the argument is written as an exponent). Unless otherwise specified, the term generally refers to the positive-valued function of a real variable, ...
. In simple terms, a convex function refers to a function whose graph is shaped like a cup
, while a
concave function
In mathematics, a concave function is the negative of a convex function. A concave function is also synonymously called concave downwards, concave down, convex upwards, convex cap, or upper convex.
Definition
A real-valued function f on an ...
'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
Mathematical optimization (alternatively spelled ''optimisation'') or mathematical programming is the selection of a best element, with regard to some criterion, from some set of available alternatives. It is generally divided into two subfi ...
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
The calculus of variations (or Variational Calculus) is a field of mathematical analysis that uses variations, which are small changes in functions
and functionals, to find maxima and minima of functionals: mappings from a set of functions t ...
. In
probability theory
Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set o ...
, a convex function applied to the
expected value
In probability theory, the expected value (also called expectation, expectancy, mathematical expectation, mean, average, or first moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean of a ...
of a
random variable
A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the p ...
is always bounded above by the expected value of the convex function of the random variable. This result, known as
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. It was proved by Jensen in 1906, building on an earlier ...
, can be used to deduce inequalities such as the
arithmetic–geometric mean inequality and
Hölder's inequality
In mathematical analysis, Hölder's inequality, named after Otto Hölder, is a fundamental inequality between integrals and an indispensable tool for the study of spaces.
:Theorem (Hölder's inequality). Let be a measure space and let with . ...
.
Definition
Let
be a
convex subset
In geometry, a subset of a Euclidean space, or more generally an affine space over the reals, is convex if, given any two points in the subset, the subset contains the whole line segment that joins them. Equivalently, a convex set or a conve ...
of a real
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but ...
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. Similarly, the argument of the function in the left hand side represents the straight line between and in or the -axis of the graph of So, this condition requires that the straight line between any pair of points on the curve of to be above or just meets the graph.
- For all and all such that :
The difference of this second condition with respect to the first condition above is that this condition does not include the intersection points (for example, and ) between the straight line passing through a pair of points on the curve of (the straight line is represented by the right hand side of this condition) and the curve of the first condition includes the intersection points as it becomes or at or or In fact, the intersection points do not need to be considered in a condition of convex using because and are always true (so not useful to be a part of a condition).
The second statement characterizing convex functions that are valued in the real line
is also the statement used to define that are valued in the
extended real number line
In mathematics, the affinely extended real number system is obtained from the real number system \R by adding two infinity elements: +\infty and -\infty, where the infinities are treated as actual numbers. It is useful in describing the algebra o ...
where such a function
is allowed to take
as a value. The first statement is not used because it permits
to take
or
as a value, in which case, if
or
respectively, then
would be undefined (because the multiplications
and
are undefined). The sum
is also undefined so a convex extended real-valued function is typically only allowed to take exactly one of
and
as a value.
The second statement can also be modified to get the definition of , where the latter is obtained by replacing
with the strict inequality
Explicitly, the map
is called if and only if for all real
and all
such that
:
A strictly convex function
is a function that the straight line between any pair of points on the curve
is above the curve
except for the intersection points between the straight line and the curve.
The function
is said to be (resp. ) if
(
multiplied by −1) is convex (resp. strictly convex).
Alternative naming
The term ''convex'' is often referred to as ''convex down'' or ''concave upward'', and the term
concave
Concave or concavity may refer to:
Science and technology
* Concave lens
* Concave mirror
Mathematics
* Concave function, the negative of a convex function
* Concave polygon, a polygon which is not convex
* Concave set
In geometry, a subset ...
is often referred as ''concave down'' or ''convex upward''. If the term "convex" is used without an "up" or "down" keyword, then it refers strictly to a cup shaped graph
. As an example,
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. It was proved by Jensen in 1906, building on an earlier ...
refers to an inequality involving a convex or convex-(up), function.
Properties
Many properties of convex functions have the same simple formulation for functions of many variables as for functions of one variable. See below the properties for the case of many variables, as some of them are not listed for functions of one variable.
Functions of one variable
* Suppose
is a function of one
real
Real may refer to:
Currencies
* Brazilian real (R$)
* Central American Republic real
* Mexican real
* Portuguese real
* Spanish real
* Spanish colonial real
Music Albums
* ''Real'' (L'Arc-en-Ciel album) (2000)
* ''Real'' (Bright album) (201 ...
variable defined on an interval, and let
(note that
is the slope of the purple line in the above drawing; the function
is
symmetric
Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definit ...
in
means that
does not change by exchanging
and
).
is convex if and only if
is
monotonically non-decreasing in
for every fixed
(or vice versa). This characterization of convexity is quite useful to prove the following results.
* A convex function
of one real variable defined on some
open interval is
continuous
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** Continuous g ...
on
admits
left and right derivatives, and these are
monotonically non-decreasing. As a consequence,
is
differentiable
In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non- vertical tangent line at each interior point i ...
at all but at most
countably many
In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbe ...
points, the set on which
is not differentiable can however still be dense. If
is closed, then
may fail to be continuous at the endpoints of
(an example is shown in the
examples section).
* A
differentiable
In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non- vertical tangent line at each interior point i ...
function of one variable is convex on an interval if and only if its
derivative
In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...
is
monotonically non-decreasing on that interval. If a function is differentiable and convex then it is also
continuously differentiable
In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non- vertical tangent line at each interior point in ...
(due to
Darboux's theorem
Darboux's theorem is a theorem in the mathematical field of differential geometry and more specifically differential forms, partially generalizing the Frobenius integration theorem. It is a foundational result in several fields, the chief amon ...
).
* A differentiable function of one variable is convex on an interval if and only if its graph lies above all of its
tangent
In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. Mo ...
s:
for all
and
in the interval.
* A twice differentiable function of one variable is convex on an interval if and only if its
second derivative
In calculus, the second derivative, or the second order derivative, of a function (mathematics), function is the derivative of the derivative of . Roughly speaking, the second derivative measures how the rate of change of a quantity is itself ...
is non-negative there; this gives a practical test for convexity. Visually, a twice differentiable convex function "curves up", without any bends the other way (
inflection point
In differential calculus and differential geometry, an inflection point, point of inflection, flex, or inflection (British English: inflexion) is a point on a smooth plane curve at which the curvature changes sign. In particular, in the case ...
s). If its second derivative is positive at all points then the function is strictly convex, but the
converse
Converse may refer to:
Mathematics and logic
* Converse (logic), the result of reversing the two parts of a definite or implicational statement
** Converse implication, the converse of a material implication
** Converse nonimplication, a logical ...
does not hold. For example, the second derivative of
is
, which is zero for
but
is strictly convex.
**This property and the above property in terms of "...its derivative is monotonically non-decreasing..." are not equal since if
is non-negative on an interval
then
is monotonically non-decreasing on
while its converse is not true, for example,
is monotonically non-decreasing on
while its derivative
is not defined at some points on
.
* If
is a convex function of one real variable, and
, then
is
superadditive In mathematics, a function f is superadditive if
f(x+y) \geq f(x) + f(y)
for all x and y in the domain of f.
Similarly, a sequence \left\, n \geq 1, is called superadditive if it satisfies the inequality
a_ \geq a_n + a_m
for all m and n.
The ...
on the
positive reals
In mathematics, the set of positive real numbers, \R_ = \left\, is the subset of those real numbers that are greater than zero. The non-negative real numbers, \R_ = \left\, also include zero. Although the symbols \R_ and \R^ are ambiguously used f ...
, that is
for positive real numbers
and
.
* A function is midpoint convex on an interval
if for all
This condition is only slightly weaker than convexity. For example, a real-valued
Lebesgue measurable function
In mathematics and in particular measure theory, a measurable function is a function between the underlying sets of two measurable spaces that preserves the structure of the spaces: the preimage of any measurable set is measurable. This is in ...
that is midpoint-convex is convex: this is a theorem of
Sierpinski. In particular, a continuous function that is midpoint convex will be convex.
Functions of several variables
* A function
extended real number
In mathematics, the affinely extended real number system is obtained from the real number system \R by adding two infinity elements: +\infty and -\infty, where the infinities are treated as actual numbers. It is useful in describing the algebra o ...
s
is convex if and only if its
epigraph is a convex set.
* A differentiable function
defined on a convex domain is convex if and only if
holds for all
in the domain.
* A twice differentiable function of several variables is convex on a convex set if and only if its
Hessian matrix
In mathematics, the Hessian matrix or Hessian is a square matrix of second-order partial derivatives of a scalar-valued function, or scalar field. It describes the local curvature of a function of many variables. The Hessian matrix was developed ...
of second
partial derivative
In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). Pa ...
s is
positive semidefinite on the interior of the convex set.
* For a convex function
the
sublevel set
In mathematics, a level set of a real-valued function of real variables is a set where the function takes on a given constant value , that is:
: L_c(f) = \left\~,
When the number of independent variables is two, a level set is cal ...
s
and
with
are convex sets. A function that satisfies this property is called a and may fail to be a convex function.
* Consequently, the set of
global minimisers of a convex function
is a convex set:
- convex.
* Any
local minimum
In mathematical analysis, the maxima and minima (the respective plurals of maximum and minimum) of a function, known collectively as extrema (the plural of extremum), are the largest and smallest value of the function, either within a given r ...
of a convex function is also a
global minimum
In mathematical analysis, the maxima and minima (the respective plurals of maximum and minimum) of a function, known collectively as extrema (the plural of extremum), are the largest and smallest value of the function, either within a given r ...
. A convex function will have at most one global minimum.
*
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. It was proved by Jensen in 1906, building on an earlier ...
applies to every convex function
. If
is a random variable taking values in the domain of
then
where
denotes the
mathematical expectation. Indeed, convex functions are exactly those that satisfies the hypothesis of
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. It was proved by Jensen in 1906, building on an earlier ...
.
* A first-order
homogeneous function
In mathematics, a homogeneous function is a function of several variables such that, if all its arguments are multiplied by a scalar, then its value is multiplied by some power of this scalar, called the degree of homogeneity, or simply the '' ...
of two positive variables
and
(that is, a function satisfying
for all positive real
) that is convex in one variable must be convex in the other variable.
Operations that preserve convexity
*
is concave if and only if
is convex.
* If
is any real number then
is convex if and only if
is convex.
* Nonnegative weighted sums:
**if
and
are all convex, then so is
In particular, the sum of two convex functions is convex.
**this property extends to infinite sums, integrals and expected values as well (provided that they exist).
* Elementwise maximum: let
be a collection of convex functions. Then
is convex. The domain of
is the collection of points where the expression is finite. Important special cases:
**If
are convex functions then so is
**
Danskin's theorem: If
is convex in
then
is convex in
even if
is not a convex set.
* Composition:
**If
and
are convex functions and
is non-decreasing over a univariate domain, then
is convex. For example, if
is convex, then so is
because
is convex and monotonically increasing.
**If
is concave and
is convex and non-increasing over a univariate domain, then
is convex.
**Convexity is invariant under affine maps: that is, if
is convex with domain
, then so is
, where
with domain
* Minimization: If
is convex in
then
is convex in
provided that
is a convex set and that
* If
is convex, then its perspective
with domain
is convex.
* Let
be a vector space.
is convex and satisfies
if and only if
for any
and any non-negative real numbers
that satisfy
Strongly convex functions
The concept of strong convexity extends and parametrizes the notion of strict convexity. A strongly convex function is also strictly convex, but not vice versa.
A differentiable function
is called strongly convex with parameter
if the following inequality holds for all points
in its domain:
or, more generally,
where
is any
inner product
In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often ...
, and
is the corresponding
norm
Naturally occurring radioactive materials (NORM) and technologically enhanced naturally occurring radioactive materials (TENORM) consist of materials, usually industrial wastes or by-products enriched with radioactive elements found in the envir ...
. Some authors, such as
refer to functions satisfying this inequality as
elliptic
In mathematics, an ellipse is a plane curve surrounding two focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special type of ellipse in ...
functions.
An equivalent condition is the following:
It is not necessary for a function to be differentiable in order to be strongly convex. A third definition
[ for a strongly convex function, with parameter is that, for all in the domain and
Notice that this definition approaches the definition for strict convexity as and is identical to the definition of a convex function when Despite this, functions exist that are strictly convex but are not strongly convex for any (see example below).
If the function is twice continuously differentiable, then it is strongly convex with parameter if and only if for all in the domain, where is the identity and is the ]Hessian matrix
In mathematics, the Hessian matrix or Hessian is a square matrix of second-order partial derivatives of a scalar-valued function, or scalar field. It describes the local curvature of a function of many variables. The Hessian matrix was developed ...
, and the inequality means that is positive semi-definite. This is equivalent to requiring that the minimum eigenvalue
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denot ...
of be at least for all If the domain is just the real line, then is just the second derivative so the condition becomes . If then this means the Hessian is positive semidefinite (or if the domain is the real line, it means that ), which implies the function is convex, and perhaps strictly convex, but not strongly convex.
Assuming still that the function is twice continuously differentiable, one can show that the lower bound of implies that it is strongly convex. Using Taylor's Theorem
In calculus, Taylor's theorem gives an approximation of a ''k''-times differentiable function around a given point by a polynomial of degree ''k'', called the ''k''th-order Taylor polynomial. For a smooth function, the Taylor polynomial is the ...
there exists
such that
Then
by the assumption about the eigenvalues, and hence we recover the second strong convexity equation above.
A function is strongly convex with parameter ''m'' if and only if the function
is convex.
The distinction between convex, strictly convex, and strongly convex can be subtle at first glance. If is twice continuously differentiable and the domain is the real line, then we can characterize it as follows:
* convex if and only if for all
* strictly convex if for all (note: this is sufficient, but not necessary).
* strongly convex if and only if for all
For example, let be strictly convex, and suppose there is a sequence of points such that . Even though , the function is not strongly convex because will become arbitrarily small.
A twice continuously differentiable function on a compact domain that satisfies for all is strongly convex. The proof of this statement follows from the extreme value theorem
In calculus, the extreme value theorem states that if a real-valued function f is continuous on the closed interval ,b/math>, then f must attain a maximum and a minimum, each at least once. That is, there exist numbers c and d in ,b/math> ...
, which states that a continuous function on a compact set has a maximum and minimum.
Strongly convex functions are in general easier to work with than convex or strictly convex functions, since they are a smaller class. Like strictly convex functions, strongly convex functions have unique minima on compact sets.
Uniformly convex functions
A uniformly convex function, with modulus , is a function that, for all in the domain and , 1
The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline o ...
satisfies
where is a function that is non-negative and vanishes only at 0. This is a generalization of the concept of strongly convex function; by taking we recover the definition of strong convexity.
It is worth noting that some authors require the modulus to be an increasing function, but this condition is not required by all authors.
Examples
Functions of one variable
* The function has , so is a convex function. It is also strongly convex (and hence strictly convex too), with strong convexity constant 2.
* The function has , so is a convex function. It is strictly convex, even though the second derivative is not strictly positive at all points. It is not strongly convex.
* The absolute value function is convex (as reflected in the triangle inequality
In mathematics, the triangle inequality states that for any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side.
This statement permits the inclusion of degenerate triangles, bu ...
), even though it does not have a derivative at the point It is not strictly convex.
* The function for is convex.
* The exponential function
The exponential function is a mathematical function denoted by f(x)=\exp(x) or e^x (where the argument is written as an exponent). Unless otherwise specified, the term generally refers to the positive-valued function of a real variable, ...
is convex. It is also strictly convex, since , but it is not strongly convex since the second derivative can be arbitrarily close to zero. More generally, the function is logarithmically convex In mathematics, a function ''f'' is logarithmically convex or superconvex if \circ f, the composition of the logarithm with ''f'', is itself a convex function.
Definition
Let be a convex subset of a real vector space, and let be a function tak ...
if is a convex function. The term "superconvex" is sometimes used instead.
* The function with domain ,1defined by for is convex; it is continuous on the open interval but not continuous at 0 and 1.
* The function has second derivative ; thus it is convex on the set where and concave
Concave or concavity may refer to:
Science and technology
* Concave lens
* Concave mirror
Mathematics
* Concave function, the negative of a convex function
* Concave polygon, a polygon which is not convex
* Concave set
In geometry, a subset ...
on the set where
* Examples of functions that are monotonically increasing
In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of ord ...
but not convex include and .
* Examples of functions that are convex but not monotonically increasing
In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of ord ...
include and .
* The function has which is greater than 0 if so is convex on the interval . It is concave on the interval .
* The function with , is convex on the interval and convex on the interval , but not convex on the interval , because of the singularity at
Functions of ''n'' variables
* LogSumExp
The LogSumExp (LSE) (also called RealSoftMax or multivariable softplus) function (mathematics), function is a smooth maximum – a smooth function, smooth approximation to the maximum function, mainly used by machine learning algorithms. It is def ...
function, also called softmax function, is a convex function.
*The function on the domain of positive-definite matrices is convex.[
* Every real-valued ]linear transformation
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pr ...
is convex but not strictly convex, since if is linear, then . This statement also holds if we replace "convex" by "concave".
* Every real-valued affine function
In Euclidean geometry, an affine transformation or affinity (from the Latin, ''affinis'', "connected with") is a geometric transformation that preserves lines and parallelism, but not necessarily Euclidean distances and angles.
More genera ...
, that is, each function of the form is simultaneously convex and concave.
* Every norm
Naturally occurring radioactive materials (NORM) and technologically enhanced naturally occurring radioactive materials (TENORM) consist of materials, usually industrial wastes or by-products enriched with radioactive elements found in the envir ...
is a convex function, by the triangle inequality
In mathematics, the triangle inequality states that for any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side.
This statement permits the inclusion of degenerate triangles, bu ...
and positive homogeneity
In mathematics, a homogeneous function is a function of several variables such that, if all its arguments are multiplied by a scalar, then its value is multiplied by some power of this scalar, called the degree of homogeneity, or simply the ''d ...
.
* The spectral radius
In mathematics, the spectral radius of a square matrix is the maximum of the absolute values of its eigenvalues. More generally, the spectral radius of a bounded linear operator is the supremum of the absolute values of the elements of its spect ...
of a nonnegative matrix
In mathematics, a nonnegative matrix, written
: \mathbf \geq 0,
is a matrix in which all the elements are equal to or greater than zero, that is,
: x_ \geq 0\qquad \forall .
A positive matrix is a matrix in which all the elements are strictly gr ...
is a convex function of its diagonal elements.[Cohen, J.E., 1981]
Convexity of the dominant eigenvalue of an essentially nonnegative matrix
Proceedings of the American Mathematical Society, 81(4), pp.657-658.
See also
* Concave function
In mathematics, a concave function is the negative of a convex function. A concave function is also synonymously called concave downwards, concave down, convex upwards, convex cap, or upper convex.
Definition
A real-valued function f on an ...
* Convex analysis
Convex analysis is the branch of mathematics devoted to the study of properties of convex functions and convex sets, often with applications in convex minimization, a subdomain of optimization theory.
Convex sets
A subset C \subseteq X of ...
* Convex conjugate
In mathematics and mathematical optimization, the convex conjugate of a function is a generalization of the Legendre transformation which applies to non-convex functions. It is also known as Legendre–Fenchel transformation, Fenchel transformatio ...
* Convex curve
In geometry, a convex curve is a plane curve that has a supporting line through each of its points. There are many other equivalent definitions of these curves, going back to Archimedes. Examples of convex curves include the convex polygons, ...
* Convex optimization
Convex optimization is a subfield of mathematical optimization that studies the problem of minimizing convex functions over convex sets (or, equivalently, maximizing concave functions over convex sets). Many classes of convex optimization prob ...
* Geodesic convexity In mathematics — specifically, in Riemannian geometry — geodesic convexity is a natural generalization of convexity for sets and functions to Riemannian manifolds. It is common to drop the prefix "geodesic" and refer simply to "convex ...
* Hahn–Banach theorem
The Hahn–Banach theorem is a central tool in functional analysis.
It allows the extension of bounded linear functionals defined on a subspace of some vector space to the whole space, and it also shows that there are "enough" continuous linear f ...
* Hermite–Hadamard inequality
In mathematics, the Hermite–Hadamard inequality, named after Charles Hermite and Jacques Hadamard and sometimes also called Hadamard's inequality, states that if a function ƒ : 'a'', ''b''nbsp;→ R is convex, then the foll ...
* Invex function
* 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. It was proved by Jensen in 1906, building on an earlier ...
* K-convex function
* Kachurovskii's theorem In mathematics, Kachurovskii's theorem is a theorem relating the convexity of a function on a Banach space to the monotonicity of its Fréchet derivative.
Statement of the theorem
Let ''K'' be a convex subset
In geometry, a subset of a Euc ...
, which relates convexity to monotonicity
In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of orde ...
of the derivative
* Karamata's inequality
* Logarithmically convex function In mathematics, a function ''f'' is logarithmically convex or superconvex if \circ f, the composition of the logarithm with ''f'', is itself a convex function.
Definition
Let be a convex subset of a real vector space, and let be a function t ...
* Pseudoconvex function
* Quasiconvex function
In mathematics, a quasiconvex function is a real number, real-valued function (mathematics), function defined on an interval (mathematics), interval or on a convex set, convex subset of a real vector space such that the inverse image of any s ...
* Subderivative
In mathematics, the subderivative, subgradient, and subdifferential generalize the derivative to convex functions which are not necessarily differentiable. Subderivatives arise in convex analysis, the study of convex functions, often in connection ...
of a convex function
Notes
References
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* Borwein, Jonathan, and Lewis, Adrian. (2000). Convex Analysis and Nonlinear Optimization. Springer.
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* Hiriart-Urruty, Jean-Baptiste, and Lemaréchal, Claude. (2004). Fundamentals of Convex analysis. Berlin: Springer.
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External links
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{{Authority control
Convex analysis
Generalized convexity
Types of functions