Jump to content

Main menu Navigation ●Main page ●Contents ●Current events ●Random article ●About Wikipedia ●Contact us ●Donate Contribute ●Help ●Learn to edit ●Community portal ●Recent changes ●Upload file

●Create account ●Log in ●Create account ● Log in Pages for logged out editors learn more ●Contributions ●Talk

List of convexity topics

Add links ●Article ●Talk ●Read ●Edit ●View history Tools Actions ●Read ●Edit ●View history General ●What links here ●Related changes ●Upload file ●Special pages ●Permanent link ●Page information ●Cite this page ●Get shortened URL ●Download QR code ●Wikidata item Print/export ●Download as PDF ●Printable version Appearance From Wikipedia, the free encyclopedia

This is a list of convexity topics, by Wikipedia page.

Alpha blending - the process of combining a translucent foreground color with a background color, thereby producing a new blended color. This is a convex combination of two colors allowing for transparency effects in computer graphics.

Barycentric coordinates - a coordinate system in which the location of a point of a simplex (a triangle, tetrahedron, etc.) is specified as the center of mass, or barycenter, of masses placed at its vertices. The coordinates are non-negative for points in the convex hull.

Borsuk's conjecture - a conjecture about the number of pieces required to cover a body with a larger diameter. Solved by Hadwiger for the case of smooth convex bodies.

Bond convexity - a measure of the non-linear relationship between price and yield duration of a bond to changes in interest rates, the second derivative of the price of the bond with respect to interest rates. A basic form of convexity in finance.

Carathéodory's theorem (convex hull) - If a point x of R^d lies in the convex hull of a set P, there is a subset of P with d+1 or fewer points such that x lies in its convex hull.

Choquet theory - an area of functional analysis and convex analysis concerned with measures with support on the extreme points of a convex set C. Roughly speaking, all vectors of C should appear as "averages" of extreme points.

Complex convexity — extends the notion of convexity to complex numbers.

Convex analysis - the branch of mathematics devoted to the study of properties of convex functions and convex sets, often with applications in convex minimization.

Convex combination - a linear combination of points where all coefficients are non-negative and sum to 1. All convex combinations are within the convex hull of the given points.

Convex and Concave - a print by Escher in which many of the structure's features can be seen as both convex shapes and concave impressions.

Convex body - a compact convex set in a Euclidean space whose interior is non-empty.

Convex conjugate - a dual of a real functional in a vector space. Can be interpreted as an encoding of the convex hull of the function's epigraph in terms of its supporting hyperplanes.

Convex curve - a plane curve that lies entirely on one side of each of its supporting lines. The interior of a closed convex curve is a convex set.

Convex function - a function in which the line segment between any two points on the graph of the function lies above the graph.

Closed convex function - a convex function all of whose sublevel sets are closed sets.
Proper convex function - a convex function whose effective domain is nonempty and it never attains minus infinity.
Concave function - the negative of a convex function.

Convex geometry - the branch of geometry studying convex sets, mainly in Euclidean space. Contains three sub-branches: general convexity, polytopes and polyhedra, and discrete geometry.

Convex hull (aka convex envelope) - the smallest convex set that contains a given set of points in Euclidean space.

Convex lens - a lens in which one or two sides is curved or bowed outwards. Light passing through the lens is converged (or focused) to a spot behind the lens.

Convex optimization - a subfield of optimization, studies the problem of minimizing convex functions over convex sets. The convexity property can make optimization in some sense "easier" than the general case - for example, any local minimum must be a global minimum.

Convex polygon - a 2-dimensional polygon whose interior is a convex set in the Euclidean plane.

Convex polytope - an n-dimensional polytope which is also a convex set in the Euclidean n-dimensional space.

Convex set - a set in Euclidean space in which contains every segment between every two of its points.

Convexity (finance) - refers to non-linearities in a financial model. When the price of an underlying variable changes, the price of an output does not change linearly, but depends on the higher-order derivatives of the modeling function. Geometrically, the model is no longer flat but curved, and the degree of curvature is called the convexity.

Duality (optimization)

Epigraph (mathematics) - for a function f : Rn→R,^{[check spelling]} the set of points lying on or above its graph

Extreme point - for a convex set S in a real vector space, a point in S which does not lie in any open line segment joining two points of S.

Fenchel conjugate

Fenchel's inequality

Fixed-point theorems in infinite-dimensional spaces, generalise the Brouwer fixed-point theorem. They have applications, for example, to the proof of existence theorems for partial differential equations

Four vertex theorem - every convex curve has at least 4 vertices.

Gift wrapping algorithm - an algorithm for computing the convex hull of a given set of points

Graham scan - a method of finding the convex hull of a finite set of points in the plane with time complexity O(n log n)

Hadwiger conjecture (combinatorial geometry) - any convex body in n-dimensional Euclidean space can be covered by 2ⁿ or fewer smaller bodies homothetic with the original body.

Hadwiger's theorem - a theorem that characterizes the valuations on convex bodies in Rⁿ.

Helly's theorem

Hyperplane - a subspace whose dimension is one less than that of its ambient space

Indifference curve

Infimal convolute

Interval (mathematics) - a set of real numbers with the property that any number that lies between two numbers in the set is also included in the set

Jarvis march

Jensen's inequality - relates the value of a convex function of an integral to the integral of the convex function

John ellipsoid - E(K) associated to a convex body Kinn-dimensional Euclidean space Rⁿ is the ellipsoid of maximal n-dimensional volume contained within K.

Lagrange multiplier - a strategy for finding the local maxima and minima of a function subject to equality constraints

Legendre transformation - an involutive transformation on the real-valued convex functions of one real variable

Locally convex topological vector space - example of topological vector spaces (TVS) that generalize normed spaces

Macbeath regions

Mahler volume - a dimensionless quantity that is associated with a centrally symmetric convex body

Minkowski's theorem - any convex set in

\mathbb {R} ^{n}

which is symmetric with respect to the origin and with volume greater than 2n d(L) contains a non-zero lattice point

Mixed volume

Mixture density

Newton polygon - a tool for understanding the behaviour of polynomials over local fields

Radon's theorem - on convex sets, that any set of d + 2 points in R^d can be partitioned into two disjoint sets whose convex hulls intersect

Separating axis theorem

Shapley–Folkman lemma - a result in convex geometry with applications in mathematical economics that describes the Minkowski addition of sets in a vector space

Shephard's problem - a geometrical question

Simplex - a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions

Simplex method - a popular algorithm for linear programming

Subdifferential - generalization of the derivative to functions which are not differentiable

Supporting hyperplane - a hyperplane meeting certain conditions

Supporting hyperplane theorem - that defines a supporting hyperplane

v t e Convex analysis and variational analysis
Basic concepts	Convex combination Convex function Convex set
Topics (list)	Choquet theory Convex geometry Convex metric space Convex optimization Duality Lagrange multiplier Legendre transformation Locally convex topological vector space Simplex
Maps	Convex conjugate Concave (Closed K- Logarithmically Proper Pseudo- Quasi-) Convex function Invex function Legendre transformation Semi-continuity Subderivative
Main results (list)	Carathéodory's theorem Ekeland's variational principle Fenchel–Moreau theorem Fenchel-Young inequality Jensen's inequality Hermite–Hadamard inequality Krein–Milman theorem Mazur's lemma Shapley–Folkman lemma Robinson–Ursescu Simons Ursescu
Sets	Convex hull (Orthogonally, Pseudo-) Convex set Effective domain Epigraph Hypograph John ellipsoid Lens Radial set/Algebraic interior Zonotope
Series	Convex series related ((cs, lcs)-closed, (cs, bcs)-complete, (lower) ideally convex, (Hx), and (Hwx))
Duality	Dual system Duality gap Strong duality Weak duality
Applications and related	Convexity in economics

Retrieved from "https://en.wikipedia.org/w/index.php?title=List_of_convexity_topics&oldid=1219308765" Categories: ●Mathematics-related lists ●Mathematical analysis ●Convex geometry Hidden categories: ●Articles with short description ●Short description is different from Wikidata ●All articles with unidentified words ●Articles with unidentified words from April 2024 ●This page was last edited on 16 April 2024, at 23:55 (UTC). ●Text is available under the Creative Commons Attribution-ShareAlike License 4.0; additional terms may apply. By using this site, you agree to the Terms of Use and Privacy Policy. Wikipedia® is a registered trademark of the Wikimedia Foundation, Inc., a non-profit organization. ●Privacy policy ●About Wikipedia ●Disclaimers ●Contact Wikipedia ●Code of Conduct ●Developers ●Statistics ●Cookie statement ●Mobile view