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Mathematics of convex functions and sets
Convex analysis is the branch of mathematics that studies convex sets, convex functions, and their applications to optimization, functional analysis, variational
Convex_analysis
In geometry, set whose intersection with every line is a single line segment
devoted to the study of properties of convex sets and convex functions is called convex analysis. Spaces in which convex sets are defined include the Euclidean
Convex_set
Real function with secant line between points above the graph itself
nonnegative matrix is a convex function of its diagonal elements. Concave function Convex analysis Convex conjugate Convex curve Convex optimization Geodesic
Convex_function
Subfield of mathematical optimization
Convex optimization is a subfield of mathematical optimization that studies the problem of minimizing convex functions over convex sets (or, equivalently
Convex_optimization
In the field of mathematics known as convex analysis, the indicator function of a set is a convex function that indicates the membership (or non-membership)
Indicator function (convex analysis)
Indicator_function_(convex_analysis)
Generalization of the Legendre transformation
mathematical optimization, the convex conjugate of a function is a generalization of the Legendre transformation which applies to non-convex functions. It is also
Convex_conjugate
Smallest convex set containing a given set
In geometry, the convex hull, convex envelope or convex closure of a shape is the smallest convex set that contains it. The convex hull may be defined
Convex_hull
Mathematical set closed under positive linear combinations
combinations with positive coefficients. It follows that convex cones are convex sets. The definition of a convex cone makes sense in a vector space over any ordered
Convex_cone
Space with topology generated by convex sets
In functional analysis and related areas of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological
Locally convex topological vector space
Locally_convex_topological_vector_space
Cone of outward normals to a convex set at a point
In convex analysis and optimization, the normal cone to a set at a point is a convex cone consisting of vectors that make a non-acute angle with every
Normal_cone_(convex_analysis)
Branch of mathematics
proof of the Poincaré conjecture. Convex analysis is the branch of analysis concerned with convex functions, convex sets, and applications to optimization
Mathematical_analysis
Branch of geometry
naturally in many areas: computational geometry, convex analysis, discrete geometry, functional analysis, geometry of numbers, integral geometry, linear
Convex_geometry
Linear combination of points where all coefficients are non-negative and sum to 1
In convex geometry and vector algebra, a convex combination is a linear combination of points (which can be vectors, scalars, or more generally points
Convex_combination
Normed vector space for which the closed unit ball is strictly convex
strictly convex space is a normed vector space (X, || ||) for which the closed unit ball is a strictly convex set. Put another way, a strictly convex space
Strictly_convex_space
\operatorname {acl} A={\overline {A}}} for every finite-dimensional convex set A. Moreover, a convex set is algebraically closed if and only if its complement is
Algebraic closure (convex analysis)
Algebraic_closure_(convex_analysis)
Theorem of convex functions
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
Jensen's_inequality
Non-empty convex set in Euclidean space
Fundamentals of Convex Analysis. doi:10.1007/978-3-642-56468-0. ISBN 978-3-540-42205-1. Rockafellar, R. Tyrrell (12 January 1997). Convex Analysis. Princeton
Convex_body
Concept in convex analysis
mathematical analysis, in particular the subfields of convex analysis and optimization, a proper convex function is an extended real-valued convex function
Proper_convex_function
theorem is a result in convex analysis named after French mathematician Jean-Jacques Moreau. It shows that sufficiently well-behaved convex functionals on Hilbert
Moreau's_theorem
Concept of mathematics in convex analysis
specifically in convex analysis, the convex compactification is a compactification which is simultaneously a convex subset in a locally convex space in functional
Convex_compactification
Terms in Maths
(2004). Convex optimization (PDF). New York: Cambridge. pp. 639–640. ISBN 978-0521833783. Rockafellar, R. Tyrrell (1997) [1970]. Convex Analysis. Princeton
Closed_convex_function
American mathematician
and related fields of analysis and combinatorics. He is the author of four major books including the landmark text "Convex Analysis" (1970), which has been
R._Tyrrell_Rockafellar
Geometric relation between the roots of a polynomial and those of its derivative
within the convex hull of the roots of P, that is the smallest convex polygon containing the roots of P. When P has a single root then this convex hull is
Gauss–Lucas_theorem
Every symmetric convex set in R^n with volume > 2^n contains a non-zero integer point
In mathematics, Minkowski's theorem is the statement that every convex set in R n {\displaystyle \mathbb {R} ^{n}} which is symmetric with respect to the
Minkowski's_theorem
Concepts in convex analysis
Dual cone and polar cone are closely related concepts in convex analysis, a branch of mathematics. The dual cone C* of a subset C in a linear space X
Dual_cone_and_polar_cone
Theorem in optimal transport
plan of an absolutely continuous probability measure is the gradient of a convex function. More precisely, if μ {\displaystyle \mu } and ν {\displaystyle
Brenier's_theorem
Upper bound on the volume of a convex body containing one lattice point
numbers, Ehrhart's volume conjecture gives an upper bound on the volume of a convex body containing only one lattice point in its interior. It is a kind of
Ehrhart's_volume_conjecture
Mathematical function
K-convex functions, first introduced by Scarf, are a special weakening of the concept of convex function which is crucial in the proof of the optimality
K-convex_function
Concept in mathematics of vector spaces
In mathematics, uniformly convex spaces (or uniformly rotund spaces) are common examples of reflexive Banach spaces. The concept of uniform convexity was
Uniformly_convex_space
In mathematics, variational analysis is the combination and extension of methods from convex optimization and the classical calculus of variations to a
Variational_analysis
function. A related concept is convexification – converting a non-convex function to a convex function. It is especially important in economics and mathematical
Concavification
Property of point sets in Euclidean spaces
\mathbb {R} ^{n}} is called a star domain (or star-convex set, star-shaped set or radially convex set) if there exists an s 0 ∈ S {\displaystyle s_{0}\in
Star_domain
Convex and balanced set
of a real or complex vector space is said to be absolutely convex or disked if it is convex and balanced (some people use the term "circled" instead of
Absolutely_convex_set
Type of plane curve
Examples of convex curves include the convex polygons, the boundaries of convex sets, and the graphs of convex functions. Important subclasses of convex curves
Convex_curve
{\displaystyle \mathbb {C} } -convex if its intersection with any complex line is contractible. In complex geometry and analysis, the notion of convexity and
Complex_convexity
Negative of a convex function
which the function value at any convex combination of elements in the domain is greater than or equal to that convex combination of those domain elements
Concave_function
Mathematical function with convex lower level sets
on a convex subset of a real vector space, such that for any real number y, the set of points on which the function value is at most y is a convex set
Quasiconvex_function
Area of functional analysis and convex analysis
an area of functional analysis and convex analysis concerned with measures which have support on the extreme points of a convex set C. Roughly speaking
Choquet_theory
Mathematical inequality about convex functions
In convex analysis, Popoviciu's inequality is an inequality about convex functions. It is similar to Jensen's inequality and was found in 1965 by Tiberiu
Popoviciu's_inequality
Branch of mathematics
such as convex bodies and normed spaces, as the dimension tends to infinity. It is at the intersection of convex geometry and functional analysis. The primary
Asymptotic_geometry
Carl Friedrich Gauss (1777–1855) is the eponym of all of the topics listed below. There are over 100 topics all named after this German mathematician and
List of things named after Carl Friedrich Gauss
List_of_things_named_after_Carl_Friedrich_Gauss
In mathematics, particularly in functional analysis and convex analysis, a convex series is a series of the form ∑ i = 1 ∞ r i x i {\displaystyle \sum
Convex_series
Mathematical transformation
real-valued functions that are convex on a real variable. Specifically, if a real-valued multivariable function is convex on one of its independent real
Legendre_transformation
Generalization of topological interior
1970]. Convex Analysis. Princeton, NJ: Princeton University Press. Theorem 6.9. ISBN 978-0-691-01586-6. Zălinescu, Constantin (30 July 2002). Convex Analysis
Relative_interior
Function in mathematical analysis
In mathematics, a Schur-convex function, also known as S-convex, isotonic function and order-preserving function is a function f : R d → R {\displaystyle
Schur-convex_function
Set of vectors in convex analysis
In mathematics, especially convex analysis, the recession cone of a set A {\displaystyle A} is a cone containing all vectors such that A {\displaystyle
Recession_cone
(functional analysis) Hahn–Banach theorem (functional analysis) Hilbert projection theorem (convex analysis) Kachurovskii's theorem (convex analysis) Kirszbraun
List_of_theorems
Mathematical theorem in convex analysis
In convex analysis, the Fenchel–Moreau theorem (named after Werner Fenchel and Jean Jacques Moreau) or Fenchel biconjugation theorem (or just biconjugation
Fenchel–Moreau_theorem
Mathematics concept
In mathematics, the lower convex envelope f ˘ {\displaystyle {\breve {f}}} of a function f {\displaystyle f} defined on an interval [ a , b ] {\displaystyle
Lower_convex_envelope
Theorem
{\displaystyle L^{\infty }(\Omega )} if and only if f {\displaystyle f} is convex. Discontinuous linear functional Renardy, Michael & Rogers, Robert C. (2004)
Tonelli's theorem (functional analysis)
Tonelli's_theorem_(functional_analysis)
Generalization of derivatives to real-valued functions
that point. Subderivatives arise in convex analysis, the study of convex functions, often in connection to convex optimization. Let f : I → R {\displaystyle
Subderivative
through the following diagram: f convex ⟹ f polyconvex ⟹ f quasiconvex ⟹ f rank-one convex {\displaystyle f{\text{ convex}}\implies f{\text{ polyconvex}}\implies
Polyconvex_function
Principle in mathematical optimization
Berlin GmbH. ISBN 978-3-8325-2503-3. Zălinescu, Constantin (2002). Convex analysis in general vector spaces. River Edge, NJ: World Scientific Publishing Co
Duality_(optimization)
Theorem in convex analysis
In convex analysis, Danskin's theorem is a theorem which provides information about the derivatives of a function of the form f ( x ) = max z ∈ Z ϕ (
Danskin's_theorem
Russian mathematician
recognized expert in convex optimization, especially in the development of efficient algorithms and numerical optimization analysis. He is currently a emeritus
Yurii_Nesterov
Violations of the convexity assumptions of elementary economics
inefficient. Non-convex economies are studied with nonsmooth analysis, which is a generalization of convex analysis. If a preference set is non-convex, then some
Non-convexity_(economics)
Theorem in convex analysis
a theorem in functional analysis that characterizes the bipolar (that is, the polar of the polar) of a set. In convex analysis, the bipolar theorem refers
Bipolar_theorem
Significant topic in economics
the tools for convex functions and their properties is called convex analysis; non-convex phenomena are studied under nonsmooth analysis. The economics
Convexity_in_economics
1090/S0273-0979-1979-14595-6. MR 0526967. Ekeland, Ivar; Temam, Roger (1999). Convex analysis and variational problems. Classics in applied mathematics. Vol. 28
Ekeland's variational principle
Ekeland's_variational_principle
modulus of convexity and the characteristic of convexity are measures of "how convex" the unit ball in a Banach space is. In some sense, the modulus of convexity
Modulus and characteristic of convexity
Modulus_and_characteristic_of_convexity
Type of mathematical function
In convex analysis, a non-negative function f: Rn → R+ is logarithmically concave (or log-concave for short) if its domain is a convex set, and if it
Logarithmically concave function
Logarithmically_concave_function
On when a space equals the closed convex hull of its extreme points
mathematical theory of functional analysis, the Krein–Milman theorem is a proposition about compact convex sets in locally convex topological vector spaces (TVSs)
Krein–Milman_theorem
Point not between two other points
In mathematics, an extreme point of a convex set S {\displaystyle S} in a real or complex vector space or affine space is a point in S {\displaystyle S}
Extreme_point
On closed convex subsets in Hilbert space
result of convex analysis that says that for every vector x {\displaystyle x} in a Hilbert space H {\displaystyle H} and every nonempty closed convex C ⊆ H
Hilbert_projection_theorem
A convex cap is a well defined structure in mathematics commonly used in convex geometry for approximating convex shapes. It is used in the construction
Convex_cap
Type of function
In convex analysis and the calculus of variations, both branches of mathematics, a pseudoconvex function is a function that behaves like a convex function
Pseudoconvex_function
Mathematical object
mathematics, a random polytope is a structure commonly used in convex analysis and the analysis of linear programs in d-dimensional Euclidean space R d {\displaystyle
Random_polytope
Statistical method
interpretations including in terms of geometry, Bayesian statistics and convex analysis. The LASSO is closely related to basis pursuit denoising. Lasso was
Lasso_(statistics)
Form of projection
used to solve non-differentiable convex optimization problems. Many interesting problems can be formulated as convex optimization problems of the form
Proximal_gradient_method
French mathematician (born 1944)
success with convex minimization methods on problems that were known to be non-convex. Ekeland's analysis explained the success of methods of convex minimization
Ivar_Ekeland
Theorem in complex analysis
M(x)=\sup _{y}|f(x+iy)|} then log M ( x ) {\displaystyle \log M(x)} is a convex function on [ a , b ] . {\displaystyle [a,b].} In other words, if x = t
Hadamard_three-lines_theorem
Solving an optimization problem with a quadratic objective function
augmented Lagrangian algorithm for solving convex quadratic optimization problems" (PDF). Journal of Convex Analysis. 12: 45–69. Archived (PDF) from the original
Quadratic_programming
Set of points touching all convex bodies of unit volume
mathematics In geometry, a Danzer set is a set of points that touches every convex body of unit volume. Ludwig Danzer asked whether it is possible for such
Danzer_set
Mathematical result in convex functions theory
theorem is a result in the theory of convex functions named after Werner Fenchel. Let f {\displaystyle f} be a proper convex function on R n {\displaystyle
Fenchel's_duality_theorem
American mathematician (1921–2008)
contributed to the fields of mathematical economics, game theory, and convex analysis. Gale graduated with a Bachelor of Arts from Swarthmore College, obtained
David_Gale
Solvability theorem for finite systems of linear inequalities
Analysis of Production and Allocation, Wiley. See Lemma 1 on page 318. Boyd, Stephen P.; Vandenberghe, Lieven (2004), "Section 5.8.3" (pdf), Convex Optimization
Farkas'_lemma
Constructions in nonsmooth analysis
things, the geometrical foundation for generalizing the convex subdifferential to non-convex functions. Of particular note is also their role in generalizing
Normal cone (variational analysis)
Normal_cone_(variational_analysis)
Algebra theorem about convex functions
as the majorization inequality, is a theorem in elementary algebra for convex and concave real-valued functions, defined on an interval of the real line
Karamata's_inequality
Italian mathematician
estimates and estimates for variational integrals using novel convex analysis, harmonic analysis, and potential theory”. In 2023, De Filippis was elected to
Cristiana_De_Filippis
German mathematician (1905–1988)
and to optimization theory. Fenchel established the basic results of convex analysis and nonlinear optimization theory which would, in time, serve as the
Werner_Fenchel
by Geoffrey Colin Shephard in 1964: if K and L are centrally symmetric convex bodies in n-dimensional Euclidean space such that whenever K and L are projected
Shephard's_problem
Region above a graph
this same purpose in the fields of convex analysis and variational analysis, in which the primary focus is on convex functions valued in [ − ∞ , ∞ ] {\displaystyle
Epigraph_(mathematics)
Region underneath a graph
Epigraph (mathematics) – Region above a graph Proper convex function – Concept in convex analysis Wikimedia Commons has media related to epigraphs und
Hypograph_(mathematics)
Property of functions which is weaker than continuity
role in convex analysis. Given a convex (extended real) function, the epigraph might not be closed. But the lower semicontinuous hull of a convex function
Semi-continuity
Type of mathematical functions
The polynomially convex hull contains the holomorphically convex hull. The domain G {\displaystyle G} is called holomorphically convex if for every compact
Function of several complex variables
Function_of_several_complex_variables
Smooth approximation to the maximum function
El Ghaoui, Laurent (2017). Optimization Models and Applications. "convex analysis - About the strictly convexity of log-sum-exp function - Mathematics
LogSumExp
Mathematical theorem
Banach space to the monotonicity of its Fréchet derivative. Let K be a convex subset of a Banach space V and let f : K → R ∪ {+∞} be an extended real-valued
Kachurovskii's_theorem
In convex analysis and mathematical optimization, the supporting functional is a generalization of the supporting hyperplane of a set. Let X be a locally
Supporting_functional
Topological set
"Separation of Convex Sets in Linear Topological Spaces" (PDF). Retrieved November 14, 2012. Nikolaĭ Kapitonovich Nikolʹskiĭ (1992). Functional analysis I: linear
Radial_set
In Euclidean space, a measure of that set's "size"
solutions to Laplace's equation Choquet theory – Area of functional analysis and convex analysis Brélot, Marcel (1967) [1960], Lectures on potential theory (Notes
Capacity_of_a_set
In convex analysis, a branch of mathematics, the effective domain extends of the domain of a function defined for functions that take values in the extended
Effective_domain
Asymptotic analysis – studies a method of describing limiting behaviour Convex analysis – studies the properties of convex functions and convex sets List
List_of_real_analysis_topics
Concept in convex optimization mathematics
Subgradient methods are convex optimization methods which use subderivatives. Originally developed by Naum Z. Shor and others in the 1960s and 1970s, subgradient
Subgradient_method
Chinese-American mathematician (1914–2010)
analysis, convex analysis and inequalities, fixed point theory, operator and matrix theory, linear and nonlinear programming, complex analysis, topology
Ky_Fan
Length of a line segment
distance is thus preferred in optimization theory, since it allows convex analysis to be used. Since squaring is a monotonic function of non-negative
Euclidean_distance
Geometric property of a pair of sets of points in Euclidean geometry
Equivalently, two sets are linearly separable precisely when their respective convex hulls are disjoint (colloquially, do not overlap). Three non-collinear points
Linear_separability
Function made from a set
− ∞ {\textstyle 0\cdot -\infty } remain undefined. In the field of convex analysis, the map p K {\textstyle p_{K}} taking on the value of ∞ {\textstyle
Minkowski_functional
Fictitious American mathematician
name Rainwater mainly in functional analysis, particularly in the geometric theory of Banach spaces and in convex functions. Rainwater's theorem is an
John_Rainwater
Subset of all points that is bounded by some given point of a dual (in a dual pairing)
functional and convex analysis, and related disciplines of mathematics, the polar set A ∘ {\displaystyle A^{\circ }} is a special convex set associated
Polar_set
Branch of mathematics
close connections to convex analysis, optimization and functional analysis and important applications in number theory. Convex geometry dates back to
Geometry
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