List of convexity topics

Short description: Wikipedia list article

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 curve that lies entirely on one side of each of its tangents. The interior of a 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 - 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, 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
• Mahler volume - a dimensionless quantity that is associated with a centrally symmetric convex body
• Minkowski's theorem - any convex set in ℝn 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 Rd 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