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Terms in Maths
t\}} is closed, then the function f {\displaystyle f} is closed. This definition is valid for any function, but most used for convex functions. A proper
Closed_convex_function
Real function with secant line between points above the graph itself
function is called convex if the line segment between any two distinct points on the graph of the function lies above or on the graph of the function
Convex_function
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,
Convex_analysis
In geometry, set whose intersection with every line is a single line segment
the function) is a convex set. Convex minimization is a subfield of optimization that studies the problem of minimizing convex functions over convex sets
Convex_set
Generalization of the Legendre transformation
optimization, the convex conjugate of a function is a generalization of the Legendre transformation which applies to non-convex functions. It is also known
Convex_conjugate
Distance from origin of tangent hyperplanes
In mathematics, the support function hA of a non-empty closed convex set A in R n {\displaystyle \mathbb {R} ^{n}} describes the (signed) distances of
Support_function
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
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
Property of functions which is weaker than continuity
in convex analysis. Given a convex (extended real) function, the epigraph might not be closed. But the lower semicontinuous hull of a convex function is
Semi-continuity
Topics referred to by the same term
Strictly convex may refer to: Strictly convex function, a function having the line between any two points above its graph Strictly convex polygon, a polygon
Strictly_convex
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
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
Class of mathematical functions
Intuitively, subharmonic functions are related to convex functions of one variable as follows. If the graph of a convex function and a line intersect at
Subharmonic_function
Type of mathematical functions
manageable condition than a holomorphically convex. The subharmonic function looks like a kind of convex function, so it was named by Levi as a pseudoconvex
Function of several complex variables
Function_of_several_complex_variables
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)
Space with topology generated by convex sets
analysis and related areas of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces
Locally convex topological vector space
Locally_convex_topological_vector_space
Mathematical set closed under positive linear combinations
Euclidean space. A convex cone is a cone that is also closed under addition, or, equivalently, a subset of a vector space that is closed under linear combinations
Convex_cone
self-concordant barrier is a particular self-concordant function, that is also a barrier function for a particular convex set. Self-concordant barriers are important
Self-concordant_function
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
Concept in mathematics of vector spaces
be replaced with the closed unit ball in the definition. Namely, a normed vector space X {\displaystyle X} is uniformly convex if and only if for every
Uniformly_convex_space
Theorem in topology
Brouwer. It states that for any continuous function f {\displaystyle f} mapping a nonempty compact convex set to itself, there is a point x 0 {\displaystyle
Brouwer_fixed-point_theorem
Theorem on extension of bounded linear functionals
1.} Every sublinear function is a convex function. On the other hand, if p : X → R {\displaystyle p:X\to \mathbb {R} } is convex with p ( 0 ) ≥ 0 , {\displaystyle
Hahn–Banach_theorem
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
All numbers between two given numbers
{\displaystyle \kappa } copies of the intervals. The concepts of convex sets and convex components are used in a proof that every totally ordered set endowed
Interval_(mathematics)
continuous convex function and for each y in X there is a unique geodesic ray δ such that δ(0) = y and, for any r > 0, the ray δ cuts each closed convex set
Busemann_function
Mathematical function having a characteristic S-shaped curve or sigmoid curve
asymptotes as x → ± ∞ {\displaystyle x\rightarrow \pm \infty } . A sigmoid function is convex for values less than a particular point, and it is concave for values
Sigmoid_function
Principle in mathematical optimization
with replacing a non-convex function with its convex closure, that is the function that has the epigraph that is the closed convex hull of the original
Duality_(optimization)
Mathematical set with an ordering
with convex sets of geometry, one uses order-convex instead of "convex". A convex sublattice of a lattice L is a sublattice of L that is also a convex set
Partially_ordered_set
Vector space with a notion of nearness
{\displaystyle X.} Closed hulls In a locally convex space, convex hulls of bounded sets are bounded. This is not true for TVSs in general. The closed convex hull of
Topological_vector_space
Strong form of uniform continuity
applications. Let F(x) be an upper semi-continuous function of x, and that F(x) is a closed, convex set for all x. Then F is one-sided Lipschitz if ( x
Lipschitz_continuity
Fixed-point theorem for set-valued functions
compact and convex subset of some Euclidean space Rn. Let φ: S → 2S be a set-valued function on S with the following properties: φ has a closed graph; φ(x)
Kakutani_fixed-point_theorem
section, closed subsets of Euclidean spaces are convex metric spaces if and only if they are convex sets. It is then natural to think of convex metric spaces
Convex_metric_space
Algorithms for solving convex optimization problems
a convex function and G is a convex set. Without loss of generality, we can assume that the objective f is a linear function. Usually, the convex set
Interior-point_method
Sums vector sets A and B by adding each vector in A to each vector in B
Minkowski inequality, the function hK+pL is again positive homogeneous and convex and hence the support function of a compact convex set. This definition is
Minkowski_addition
On the existence of a continuous selection of a multivalued map from a paracompact space
{\displaystyle F\colon X\to Y} be a lower hemicontinuous set-valued function with nonempty convex closed values. Then there exists a continuous selection f : X →
Michael_selection_theorem
Operation on the subsets of a set
replace "closed sets" by "closed elements" and "intersection" by "greatest lower bound". Operations and (partial) multivariate function are examples of such
Closure_(mathematics)
Point where function's value is zero
In analysis and geometry, any closed subset of R n {\displaystyle \mathbb {R} ^{n}} is the zero set of a smooth function defined on all of R n {\displaystyle
Zero_of_a_function
Form of projection
simultaneously several convex constraints. Let f i {\displaystyle f_{i}} be the indicator function of non-empty closed convex set C i {\displaystyle C_{i}}
Proximal_gradient_method
Mathematical transformation
transformation on real-valued functions that are convex on a real variable. Specifically, if a real-valued multivariable function is convex on one of its independent
Legendre_transformation
Four-dimensional analogues of the regular polyhedra in three dimensions
{\frac {\pi }{q}}} to ensure that the cells meet to form a closed 3-surface. The six convex and ten star polytopes described are the only solutions to
Regular_4-polytope
Complement of an open subset
convex analysis, closedness is commonly expressed through epigraphs. A convex function is called closed when its epigraph is a closed set. This condition
Closed_set
Method to solve optimization problems
of a linear objective function, subject to linear equality and linear inequality constraints. Its feasible region is a convex polytope, which is a set
Linear_programming
Construct in functional analysis
origin and every convex neighborhood of the origin contains a balanced convex neighborhood of the origin (even if the TVS is not locally convex). This neighborhood
Balanced_set
Measure of difference between two points
measure of difference between two points, defined in terms of a strictly convex function; they form an important class of divergences. When the points are interpreted
Bregman_divergence
Theorems generalizing the Brouwer fixed-point theorem
nonempty closed bounded convex set in a uniformly convex Banach space. Then any non-expansive function f : K → K has a fixed point. (A function f {\displaystyle
Fixed-point theorems in infinite-dimensional spaces
Fixed-point_theorems_in_infinite-dimensional_spaces
The area cut off by a secant of a smooth convex oval is not an algebraic function
ovals states that the area cut off by a secant of a smooth convex oval is not an algebraic function of the secant. Isaac Newton stated it as lemma 28 of section VI
Newton's_theorem_about_ovals
Extension of the factorial function
is the unique interpolating function for the factorial, defined over the positive reals, which is logarithmically convex, meaning that y = log f ( x
Gamma_function
Function reducing distance between all points
is closed under convex combinations, but not compositions. This class includes proximal mappings of proper, convex, lower-semicontinuous functions, hence
Contraction_mapping
the graph. Closed convex function - a convex function all of whose sublevel sets are closed sets. Proper convex function - a convex function whose effective
List_of_convexity_topics
Provides conditions for a parametric optimization problem to have continuous solutions
and C {\displaystyle C} is convex-valued, then C ∗ {\displaystyle C^{*}} is single-valued, and thus is a continuous function rather than a correspondence
Maximum_theorem
Generalized function whose value is zero everywhere except at zero
Moreover, the convex hull of the image of X under this embedding is dense in the space of probability measures on X. The delta function satisfies the
Dirac_delta_function
Theorem relating continuity to graphs
In mathematics, the closed graph theorem may refer to one of several basic results characterizing continuous functions in terms of their graphs. Each gives
Closed_graph_theorem
Area of functional analysis and convex analysis
Euclidean space, a bounded, closed convex set C is the convex hull of its extreme point set E, so that any c in C is a (finite) convex combination of points
Choquet_theory
Relation among continuous functions
In mathematical analysis, a family of functions is equicontinuous if all the functions are continuous and they have equal variation over a given neighbourhood
Equicontinuity
On the existence of hyperplanes separating disjoint convex sets
convex sets in n-dimensional Euclidean space. There are several rather similar versions. In one version of the theorem, if both these sets are closed
Hyperplane_separation_theorem
Pentagon with all sides equal but the angles may not be equal
degrees). Four intersecting equal circles arranged in a closed chain are sufficient to determine a convex equilateral pentagon. Each circle's center is one
Equilateral_pentagon
Approximation to shape of a point cloud
generalization of the concept of the convex hull, i.e. every convex hull is an alpha-shape but not every alpha shape is a convex hull. For each real number α
Alpha_shape
of its indicator function. By induction, it is easy to show that independent of dimension, the Euler measure of a closed bounded convex polyhedron always
Euler_measure
On points of extreme curvature in curves
Thus, the evolute of any smooth closed curve has at least four cusps. The four-vertex theorem was first proved for convex curves (i.e. curves with strictly
Four_vertex_theorem
Concept in mathematical optimization
variable chosen from a convex subset of R n {\displaystyle \mathbb {R} ^{n}} , f {\displaystyle f} is the objective or utility function, g i ( i = 1 , …
Karush–Kuhn–Tucker_conditions
Existence of geodesic circles on surfaces
simple closed geodesics (i.e. three embedded geodesic circles). The result can also be extended to quasigeodesics on a convex polyhedron, and to closed geodesics
Theorem of the three geodesics
Theorem_of_the_three_geodesics
Subfield of convex optimization
of convex optimization that studies problems consisting of minimizing a convex function over the intersection of an affine subspace and a convex cone
Conic_optimization
Complex-differentiable (mathematical) function
a locally convex topological vector space, with the seminorms being the suprema on compact subsets. From a geometric perspective, a function f {\displaystyle
Holomorphic_function
Normed vector space that is complete
reflexive spaces to certain optimization problems. For example, every convex continuous function on the unit ball B {\displaystyle B} of a reflexive space attains
Banach_space
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 ( x
Danskin's_theorem
Volume space bounded by a sphere
never compact. However, a ball in a normed vector space will always be convex as a consequence of the triangle inequality. A subset of a metric space
Ball_(mathematics)
with replacing a non-convex function with its convex closure, that is the function that has the epigraph that is the closed convex hull of the original
Duality_gap
Set of probability measures
probability measures. A credal set is often assumed or constructed to be a closed convex set. It is intended to express uncertainty or doubt about the probability
Credal_set
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)
logarithmically convex sequence M {\displaystyle M} satisfies: C n M {\displaystyle C_{n}^{M}} is a ring. In particular it is closed under multiplication
Quasi-analytic_function
Region above a graph
these functions. Epigraphs serve this same purpose in the fields of convex analysis and variational analysis, in which the primary focus is on convex functions
Epigraph_(mathematics)
Sums of sets of vectors are nearly convex
are sums of many functions. In probability, it can be used to prove a law of large numbers for random sets. A set is said to be convex if every line segment
Shapley–Folkman_lemma
Type of vector space in math
space is a uniformly convex Banach space. This subsection employs the Hilbert projection theorem. If C is a non-empty closed convex subset of a Hilbert
Hilbert_space
Geometric line segment whose endpoints lie on a circular arc
Ptolemy's table of chords Holditch's theorem, for a chord rotating in a convex closed curve Circle graph Exsecant and excosecant Versine and haversine – (
Chord_(geometry)
Minimal superset that intersects each axis-parallel line in an interval
convex hull of K if and only if each of the closed axis-aligned orthants having p as apex has a nonempty intersection with K. The orthogonal convex hull
Orthogonal_convex_hull
Function spaces generalizing finite-dimensional p norm spaces
that the function F : [ 0 , ∞ ) → R {\displaystyle F:[0,\infty )\to \mathbb {R} } defined by F ( t ) = t p {\displaystyle F(t)=t^{p}} is convex, which by
Lp_space
Type of mathematical plane curve
support function of any strictly convex set is its boundary, parameterized by the angle of its supporting lines. When a convex set is not strictly convex (it
Hedgehog_(geometry)
Theorem in real analysis
is used to prove, the mean value theorem. If a real function f is continuous on a proper closed interval [a, b], differentiable on the open interval
Rolle's_theorem
American mathematician
2013.03.001. Convex analysis (cf. Werner Fenchel) Convex function Characteristic function (convex analysis) Closed convex function Convex conjugate Epigraph
R._Tyrrell_Rockafellar
Special function in the physical sciences
mathematics, the Airy function (or Airy function of the first kind) A i ( x ) {\displaystyle \mathbf {Ai({\boldsymbol {x}})} } is a special function named after
Airy_function
Theorems connecting continuity to closure of graphs
Closed Graph Theorem—Also, a closed linear map from a locally convex ultrabarrelled space into a complete pseudometrizable TVS is continuous. Closed Graph
Closed graph theorem (functional analysis)
Closed_graph_theorem_(functional_analysis)
Mathematical method
of whose values are compact and convex. If graph(Φ) is closed, then for every ε > 0 there exists a continuous function f : X → Y with graph(f) ⊂ [graph(Φ)]ε
Selection_theorem
Integral expressing the amount of overlap of one function as it is shifted over another
are μ and ν. In convex analysis, the infimal convolution of proper (not identically + ∞ {\displaystyle +\infty } ) convex functions f 1 , … , f m {\displaystyle
Convolution
Set-to-real map with diminishing returns
\sum _{S}\alpha _{S}=1,\alpha _{S}\geq 0\right)} . The convex closure of any set function is convex over [ 0 , 1 ] n {\displaystyle [0,1]^{n}} . Consider
Submodular_set_function
Function in mathematical optimization
operator is an operator associated with a proper, lower semi-continuous convex function f {\displaystyle f} from a Hilbert space X {\displaystyle {\mathcal
Proximal_operator
Economic Model
general equilibrium model. It posits that under certain economic assumptions (convex preferences, perfect competition, and demand independence), there must be
Arrow–Debreu_model
Property of a planar simple closed curve
vertex of the convex hull is chosen, one can then apply the formula using the previous and next vertices, even if those are not on the convex hull, as there
Curve_orientation
Semicontinuity for set-valued functions
{\displaystyle \Gamma :A\to P\left(\mathbb {R} ^{n}\right)} is a set-valued function with convex values and open upper sections, then Γ {\displaystyle \Gamma } has
Hemicontinuity
caveat: many terms in Riemannian and metric geometry, such as convex function, convex set and others, do not have exactly the same meaning as in general
Glossary of Riemannian and metric geometry
Glossary_of_Riemannian_and_metric_geometry
French mathematician (born 1944)
of methods of convex minimization on large problems that appeared to be non-convex. In many optimization problems, the objective function f are separable
Ivar_Ekeland
Significant topic in economics
mathematics which supplies the tools for convex functions and their properties is called convex analysis; non-convex phenomena are studied under nonsmooth
Convexity_in_economics
octonions, sedenions, trigintaduonions etc.) p-adic function: a function whose domain is p-adic. Convex function: line segment between any two points on the graph
List_of_types_of_functions
Fourier transform of the probability density function
distributions are available. The set of all characteristic functions is closed under certain operations: A convex linear combination ∑ n a n φ n ( t ) {\textstyle
Characteristic function (probability theory)
Characteristic_function_(probability_theory)
Sum of terms, each multiplied with a scalar
are closed under convex combination (they form a convex set), but not conical or affine combinations (or linear), and positive measures are closed under
Linear_combination
Soviet and Russian mathematician
theorem for a closed convex surface with the Gaussian curvature given as a function of a unit normal under some natural condition on this function; the open
Aleksei_Pogorelov
Mathematical function
need not be positive definite. Seminorms are intimately connected with convex sets: every seminorm is the Minkowski functional of some absorbing disk
Seminorm
intersection of two closed convex polygons is again a closed convex polygon, and the closure of the convex hull of their union is also a closed convex polygon. It
Bounded_lattice
Property of point sets in Euclidean spaces
{\displaystyle \mathbb {R} ^{n}} ), the convex hull of { x , y } {\displaystyle \{x,y\}} is called the closed interval with endpoints x {\displaystyle
Star_domain
Probability distribution
Pedersen also proved many properties of the median, showing that it is a convex function of α, and that the asymptotic behavior near α = 0 {\displaystyle \alpha
Gamma_distribution
Hyperplane in geometry
is a closed set with nonempty interior such that every point on the boundary has a supporting hyperplane, then S {\displaystyle S} is a convex set, and
Supporting_hyperplane
CLOSED CONVEX-FUNCTION
CLOSED CONVEX-FUNCTION
Surname or Lastname
English
English : topographic name for someone who lived by an enclosure of some sort, such as a courtyard set back from the main street or a farmyard, from Middle English clos(e) (Old French clos, from Late Latin clausum, past participle of claudere ‘to close’).English : from Middle English clos(e) ‘secret’, applied as a nickname for a reserved or secretive person.Dutch : variant of Claeys.Altered spelling of German Klose.
Female
English
Old English flower name, CLOVER means simply "clover."
Male
English
Anglicized form of Irish Gaelic Conláed, CONLEY means "purifying fire."
Surname or Lastname
English
English : from Old French covine ‘fraud’, ‘deceit’, hence a derogatory nickname for a trickster.English : habitational name from a place in Staffordshire named Coven ‘(place) at the huts or shelters (Old English cofa, dative plural cofum)’.
Surname or Lastname
English
English : from Middle English cony ‘rabbit’ (a back-formation from conies, from Old French conis, plural of conil), a nickname for someone thought to resemble a rabbit in some way or a metonymic occupational name for a dealer in rabbits or rabbit skins.
Girl/Female
American, Anglo, Australian, British, Christian, English, Jamaican, Portuguese
Clover; Flower Name; Fortunate; Mind; Heart; Spirit
Girl/Female
Tamil
Nimeelitha | நீமிலீதா
Closed
Nimeelitha | நீமிலீதா
Male
English
Variant spelling of English Connor, CONNER means "hound-lover."
Surname or Lastname
English
English : variant spelling of Close.Americanized spelling of German Klaus.
Male
English
Anglicized form of Hebrew Kesed, CHESED means "increase." In the bible, this is the name of the 4th son of Nahor.
Surname or Lastname
Italian
Italian : from the title of rank conte ‘count’ (from Latin comes, genitive comitis ‘companion’). Probably in this sense (and the Late Latin sense of ‘traveling companion’), it was a medieval personal name; as a title it was no doubt applied ironically as a nickname for someone with airs and graces or simply for someone who worked in the service of a count.English : variant of Count, cognate with 1.French : nickname for someone in the service of a count or for someone who behaved pretentiously, from Old French conte, cunte ‘count’ (of the same derivation as 1).French (Conté) : variant of Comté (see Comte).
Boy/Male
British, Christian, English
Wagoner; To Convey
Surname or Lastname
English
English : variant of Close 1.German : variant of Kloss.
Surname or Lastname
English
English : metathesized form of the occupational name Coyner.English : possibly an occupational name for a dealer in rabbits or rabbit skins, from an agent derivative of Middle English cony ‘rabbit’ (see Coney).
Girl/Female
Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Sanskrit, Sindhi, Tamil, Telugu
Close; Clove
Surname or Lastname
Spanish and Portuguese
Spanish and Portuguese : nickname from the title of rank conde ‘count’, a derivative of Latin comes, comitis ‘companion’.English : unexplained.
Surname or Lastname
English
English : habitational name from a place named Cove, examples of which are found in Devon, Hampshire, and Suffolk, from Old English cofa ‘cove’, ‘bay’, ‘inlet’, also ‘shelter’, ‘hut’, or a topographic name with the same meaning.
Surname or Lastname
English (Leicestershire)
English (Leicestershire) : variant of Culver.
Girl/Female
Anglo Saxon English
Clover.
Girl/Female
Hindu
Closed
CLOSED CONVEX-FUNCTION
CLOSED CONVEX-FUNCTION
Boy/Male
British, English
Strong Warrior
Girl/Female
Muslim/Islamic
Deep-rooted firmly established
Boy/Male
Gujarati, Hindu, Indian, Kannada, Telugu
Pure; More Meaningful
Girl/Female
American, Australian, British, Christian, Danish, English, Finnish, French, German, Greek, Hebrew, Indian, Jamaican, Swedish, Swiss
From Doris; Dorian Woman; Woman of the Sea; Gift; Gift from God; Name of a Place
Boy/Male
American, Hebrew, Hindu, Indian
Form of Joseph; God Adds; Son of Jacob and Rachel
Girl/Female
Hebrew, Hindu, Indian, Swiss
Nice
Boy/Male
Arabic, Indian, Muslim, Tamil
Honourable Judge; One who Judges Fairly; Lord of Origen; Lord of Rain
Boy/Male
Hindu, Indian, Tamil, Telugu
Lord Vishnu
Girl/Female
Hindu, Indian, Traditional
Fame
Surname or Lastname
English
English : habitational name from places in Northamptonshire and Staffordshire, so named from the Old English personal name Ecca + tūn ‘settlement’, ‘enclosure’.
CLOSED CONVEX-FUNCTION
CLOSED CONVEX-FUNCTION
CLOSED CONVEX-FUNCTION
CLOSED CONVEX-FUNCTION
CLOSED CONVEX-FUNCTION
a.
Convex on one side, and flat on the other; plano-convex.
n.
One who, or that which, closes; specifically, a boot closer. See under Boot.
a.
Convex on both sides; as, a biconvex lens.
dv.
In a convex form; convexly.
v. t.
To impart or communicate; as, to convey an impression; to convey information.
v. t.
Shut fast; closed; tight; as, a close box.
adv.
In a convex form; as, a body convexly shaped.
a.
Convex on both sides; double convex. See under Convex, a.
v. t.
Narrow; confined; as, a close alley; close quarters.
adv.
In a close manner.
imp. & p. p.
of Close
v. t.
To make into a closet for a secret interview.
a.
Plane or flat on one side, and convex on the other; as, a plano-convex lens. See Convex, and Lens.
v. t.
To make close.
n.
A convex body or surface.
a.
Firmly barred or closed.
adv.
Close; closely.
a.
Made convex; protuberant in a spherical form.
v. t.
To context.
v. t.
To accompany; to convoy.