List of convexity topics

Last updated April 17, 2024

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 K in n-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

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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 region is a subset that intersects every line into a single line segment . For example, a solid cube is a convex set, but anything that is hollow or has an indent, for example, a crescent shape, is not convex.

<span class="mw-page-title-main">Simplex</span> Multi-dimensional generalization of triangle

In geometry, a simplex is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. The simplex is so-named because it represents the simplest possible polytope in any given dimension. For example,

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 either as the intersection of all convex sets containing a given subset of a Euclidean space, or equivalently as the set of all convex combinations of points in the subset. For a bounded subset of the plane, the convex hull may be visualized as the shape enclosed by a rubber band stretched around the subset.

In geometry, a hyperplane is a generalization of a two-dimensional plane in three-dimensional space to mathematical spaces of arbitrary dimension. Like a plane in space, a hyperplane is a flat hypersurface, a subspace whose dimension is one less than that of the ambient space. Two lower-dimensional examples of hyperplanes are one-dimensional lines in a plane and zero-dimensional points on a line.

Geometry of numbers is the part of number theory which uses geometry for the study of algebraic numbers. Typically, a ring of algebraic integers is viewed as a lattice in $and the study of these lattices provides fundamental information on algebraic numbers. The geometry of numbers was initiated by Hermann Minkowski.$

Discrete geometry and combinatorial geometry are branches of geometry that study combinatorial properties and constructive methods of discrete geometric objects. Most questions in discrete geometry involve finite or discrete sets of basic geometric objects, such as points, lines, planes, circles, spheres, polygons, and so forth. The subject focuses on the combinatorial properties of these objects, such as how they intersect one another, or how they may be arranged to cover a larger object.

In geometry, the Minkowski sum of two sets of position vectors A and B in Euclidean space is formed by adding each vector in A to each vector in B:

<span class="mw-page-title-main">Convex polygon</span> Polygon that is the boundary of a convex set

In geometry, a convex polygon is a polygon that is the boundary of a convex set. This means that the line segment between two points of the polygon is contained in the union of the interior and the boundary of the polygon. In particular, it is a simple polygon. Equivalently, a polygon is convex if every line that does not contain any edge intersects the polygon in at most two points.

In geometry, Radon's theorem on convex sets, published by Johann Radon in 1921, states that:

Any set of d + 2 points in R^d can be partitioned into two sets whose convex hulls intersect.

In mathematics, the real coordinate space or real coordinate n-space, of dimension $n$ , denoted $R n$ or , is the set of all ordered $n$ -tuples of real numbers, that is the set of all sequences of $n$ real numbers, also known as coordinate vectors. Special cases are called the real line $R 1$ , the real coordinate plane $R 2$ , and the real coordinate three-dimensional space $R 3$ . With component-wise addition and scalar multiplication, it is a real vector space.

<span class="mw-page-title-main">Convex polytope</span> Convex hull of a finite set of points in a Euclidean space

A convex polytope is a special case of a polytope, having the additional property that it is also a convex set contained in the $-dimensional Euclidean space . Most texts use the term "polytope" for a bounded convex polytope, and the word "polyhedron" for the more general, possibly unbounded object. Others allow polytopes to be unbounded. The terms "bounded/unbounded convex polytope" will be used below whenever the boundedness is critical to the discussed issue. Yet other texts identify a convex polytope with its boundary.$

In linear algebra, a cone—sometimes called a linear cone for distinguishing it from other sorts of cones—is a subset of a vector space that is closed under positive scalar multiplication; that is, $C$ is a cone if $implies for every positive scalar s . A cone need not be convex, or even look like a cone in Euclidean space.$

In mathematics, Choquet theory, named after Gustave Choquet, is 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, every vector of C should appear as a weighted average of extreme points, a concept made more precise by generalizing the notion of weighted average from a convex combination to an integral taken over the set E of extreme points. Here C is a subset of a real vector space V, and the main thrust of the theory is to treat the cases where V is an infinite-dimensional topological vector space along lines similar to the finite-dimensional case. The main concerns of Gustave Choquet were in potential theory. Choquet theory has become a general paradigm, particularly for treating convex cones as determined by their extreme rays, and so for many different notions of positivity in mathematics.

In geometry, a supporting hyperplane of a set $in Euclidean space is a hyperplane that has both of the following two properties:$

In mathematics, convex geometry is the branch of geometry studying convex sets, mainly in Euclidean space. Convex sets occur naturally in many areas: computational geometry, convex analysis, discrete geometry, functional analysis, geometry of numbers, integral geometry, linear programming, probability theory, game theory, etc.

In geometry, the hyperplane separation theorem is a theorem about disjoint 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 and at least one of them is compact, then there is a hyperplane in between them and even two parallel hyperplanes in between them separated by a gap. In another version, if both disjoint convex sets are open, then there is a hyperplane in between them, but not necessarily any gap. An axis which is orthogonal to a separating hyperplane is a separating axis, because the orthogonal projections of the convex bodies onto the axis are disjoint.

Geometry is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space. Geometry is one of the oldest mathematical sciences.

In geometry and polyhedral combinatorics, an integral polytope is a convex polytope whose vertices all have integer Cartesian coordinates. That is, it is a polytope that equals the convex hull of its integer points. Integral polytopes are also called lattice polytopes or Z-polytopes. The special cases of two- and three-dimensional integral polytopes may be called polygons or polyhedra instead of polytopes, respectively.

In geometry, the moment curve is an algebraic curve in d-dimensional Euclidean space given by the set of points with Cartesian coordinates of the form

In geometry, it is an unsolved conjecture of Hugo Hadwiger that every simplex can be dissected into orthoschemes, using a number of orthoschemes bounded by a function of the dimension of the simplex. If true, then more generally every convex polytope could be dissected into orthoschemes.

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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