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Convex combination

A convex combination is a linear combination of vectors or points in a real vector space where the coefficients, known as weights, are non-negative real numbers that sum to one, resulting in a point that lies within the convex hull of the original points.^[1] Formally, for points x_1, x_2, \dots, x_k and weights \lambda_1, \lambda_2, \dots, \lambda_k \geq 0 with \sum_{i=1}^k \lambda_i = 1, the convex combination is \sum_{i=1}^k \lambda_i x_i.^[2] This concept generalizes the notion of a weighted average, ensuring that the result remains "between" the points in a geometric sense, and it forms the foundation for understanding mixtures and interpolations in finite dimensions.^[3] In convex analysis, convex combinations are pivotal for defining convex sets, which are subsets of a vector space that contain all convex combinations of any finite number of their points, thereby including all line segments connecting pairs of elements.^[2] A set is convex if and only if it is closed under such operations, with examples including the entire space \mathbb{R}^n, the non-negative orthant, Euclidean balls, and intersections of half-spaces.^[4] This property ensures that convex sets exhibit desirable geometric stability, such as being connected and "rounded" without indentations, which is crucial for theoretical developments in geometry and optimization.^[2] Convex combinations extend naturally to the theory of convex functions, where Jensen's inequality states that for a convex function f and weights \lambda_i \geq 0 summing to one, f\left(\sum_{i=1}^k \lambda_i x_i\right) \leq \sum_{i=1}^k \lambda_i f(x_i), providing a bridge between convexity in domains and function behavior.^[2] In applications, they underpin convex optimization problems, where the feasible region defined by convex combinations guarantees that any local optimum is global, enabling efficient algorithms for solving large-scale issues in fields like machine learning, operations research, and network routing.^[5] For instance, in transportation and data networks, convex combinations model resource allocations and path mixtures to achieve optimal flows.^[5]

Fundamentals

Definition

A convex combination is a linear combination of points in a vector space in which all coefficients are non-negative real numbers that sum to one.^[6] For a finite collection of points x_1, \dots, x_n in a real vector space V, the general form of such a combination is \sum_{i=1}^n \lambda_i x_i, where each \lambda_i \geq 0 and \sum_{i=1}^n \lambda_i = 1.^[7] Unlike a general linear combination, which permits arbitrary real coefficients without restrictions on sign or magnitude, a convex combination imposes the constraints of non-negativity and normalization to ensure the result lies within the "weighted average" structure specific to convex geometry.^[3] In more abstract settings, such as Banach spaces, the notion extends to infinite sums \sum_{i=1}^\infty \lambda_i x_i under conditions ensuring convergence, where the \lambda_i \geq 0 form a summable sequence with total sum 1 and the series converges in the space's norm.^[8]

Notation

A convex combination of a finite collection of points x_1, \dots, x_k in a real vector space is denoted by

\sum_{i=1}^k \lambda_i x_i,

where the coefficients \lambda_i are real numbers satisfying \lambda_i \geq 0 for each i and \sum_{i=1}^k \lambda_i = [1](/page/1). The set of all such convex combinations taken over points in a given set S is called the convex hull of S and is denoted \operatorname{conv}(S). The coefficients \lambda_i in this notation are termed the weights of the combination and, in the specific case where the points x_1, \dots, x_k form the vertices of a simplex, they coincide with the barycentric coordinates of the resulting point relative to that simplex.^[9] While the standard formulation restricts convex combinations to finite index sets, extensions to countable index sets involve infinite series \sum_{i=1}^\infty \lambda_i x_i with \lambda_i \geq 0 and \sum_{i=1}^\infty \lambda_i = 1, provided the series converges; for continuous index sets, the analogous construction uses integrals of the form \int_S x \, d\mu(x), where \mu is a probability measure supported on S.^[10] By convention, the convex hull of the empty set \emptyset is \emptyset itself, as no points are available for combination.^[11] In vector space contexts, the notion of an empty convex combination is sometimes informally associated with the zero vector, though this aligns more closely with affine combinations and is not part of the standard definition for convex ones.^[11]

Properties

Algebraic Properties

Convex combinations exhibit several key algebraic properties that facilitate their manipulation in vector spaces. One fundamental property is closure under further convex combinations: if a set C consists of all convex combinations of a fixed collection of points \{x_i\}, then C is closed under the operation of taking convex combinations of its elements. This means that if y_1, y_2, \dots, y_m \in C with weights \mu_j \geq 0 and \sum_{j=1}^m \mu_j = 1, then \sum_{j=1}^m \mu_j y_j \in C.^[12]^[13] This closure arises from the associative nature of linear combinations, allowing the regrouping of coefficients when combining multiple convex combinations. Specifically, suppose y_j = \sum_{i=1}^n \lambda_{ji} x_i for each j, where \lambda_{ji} \geq 0 and \sum_{i=1}^n \lambda_{ji} = 1. Then the combined expression is

\sum_{j=1}^m \mu_j y_j = \sum_{j=1}^m \mu_j \left( \sum_{i=1}^n \lambda_{ji} x_i \right) = \sum_{i=1}^n \left( \sum_{j=1}^m \mu_j \lambda_{ji} \right) x_i,

where the new coefficients \nu_i = \sum_{j=1}^m \mu_j \lambda_{ji} satisfy \nu_i \geq 0 and \sum_{i=1}^n \nu_i = 1, confirming that the result is again a convex combination of the original points.^[14]^[11] Another important algebraic bound is provided by Carathéodory's theorem, which states that in an n-dimensional Euclidean space, any point in the convex hull of a set can be expressed as a convex combination of at most n+1 points from the set. This limits the sparsity of representations in convex combinations, ensuring computational efficiency in algebraic manipulations.^[15]^[14] Finally, fixing a set of points \{x_1, \dots, x_k\}, the mapping from coefficients \lambda = (\lambda_1, \dots, \lambda_k) with \sum \lambda_i = 1 and \lambda_i \geq 0 to the convex combination \sum \lambda_i x_i is an affine function of \lambda. This linearity in the coefficients (affine due to the normalization constraint) underpins many algebraic derivations involving convex combinations.^[16]^[17]

Geometric Interpretation

Geometrically, a convex combination of two points in Euclidean space lies on the line segment connecting them, representing all possible weighted averages where the weights are non-negative and sum to one. For instance, the point \lambda x + (1 - \lambda) y for $0 \leq \lambda \leq 1 traces the entire segment from y to x. This extends to multiple points: the set of all convex combinations of k points forms the convex hull of those points, which is a simplex in \mathbb{R}^{d} when the points are affinely independent, filling the interior and boundary of the polytope they span.^[18] This construction admits a physical interpretation as the barycenter, or center of mass, of the points with masses proportional to the coefficients \lambda_i. If masses \lambda_i m (for total mass m = \sum \lambda_i = 1) are placed at positions x_i, the equilibrium position is the convex combination \sum \lambda_i x_i, emphasizing the "balancing" role of non-negative weights in maintaining the point within the spanned region.^[19] Convex combinations preserve the structure of convex sets: the result of taking a convex combination of points each from distinct convex sets lies within the convex hull of the union of those sets. For two convex sets A and B, the set \{ \lambda a + (1 - \lambda) b \mid a \in A, b \in B, 0 \leq \lambda \leq 1 \} is contained in \operatorname{conv}(A \cup B), illustrating how such operations generate the smallest convex set enclosing the originals; in two dimensions, this can be visualized as the region between line segments joining boundary points of A and B.^[19] In particular, convex combinations parameterize all points within standard simplices, such as the probability simplex \Delta^{n-1} = \{ x \in \mathbb{R}^n \mid x_i \geq 0, \sum x_i = 1 \}, which is the convex hull of the standard basis vectors e_1, \dots, e_n. Any point in \Delta^{n-1} is uniquely expressible as \sum_{i=1}^n \lambda_i e_i with \lambda_i \geq 0 and \sum \lambda_i = 1, providing a coordinate system for the simplex's interior.^[19]

Examples

Vector Spaces

In finite-dimensional vector spaces such as \mathbb{R}^n, convex combinations provide a concrete way to generate points within the convex hull of a given set of vectors. Consider two points in \mathbb{R}^2, namely (0,0) and (1,1). A convex combination of these points takes the form \lambda (1,1) + (1-\lambda)(0,0) = (\lambda, \lambda) where \lambda \in [0,1]. This parametrization traces out the line segment connecting the two points, illustrating how convex combinations fill the interval between endpoints.^[6] For multiple points, the concept extends naturally. Take the vertices of a triangle in \mathbb{R}^2: (0,0), (1,0), and (0,1). An interior point such as (0.3, 0.3) can be expressed as the convex combination $0.4(0,0) + 0.3(1,0) + 0.3(0,1), where the weights $0.4, $0.3, and $0.3 are nonnegative and sum to 1. Such combinations densely fill the triangular region, forming its convex hull.^[6] To determine the weights for a given point y in the convex hull of points x_1, \dots, x_k \in \mathbb{R}^n, one solves the linear system \sum_{i=1}^k \lambda_i x_i = y subject to \sum_{i=1}^k \lambda_i = 1 and \lambda_i \geq 0 for all i. This is a standard feasibility problem in linear programming, equivalent to checking membership in the convex hull; if a solution exists, the \lambda_i are the barycentric coordinates relative to the points. For the triangle example above with y = (0.3, 0.3), the system yields \lambda_1 = 0.4, \lambda_2 = 0.3, \lambda_3 = 0.3, confirming the representation.^[6] Degeneracy arises when the points are collinear, reducing the dimension of the convex hull. In this case, convex combinations of k > 2 collinear points lie solely on the line segment between the extreme endpoints, rather than spanning a higher-dimensional set; the weights for interior points can still be computed via the system, but redundant points do not contribute uniquely to the hull.^[6]

Probability Measures

In probability theory, a convex combination arises naturally as the expected value of a random variable. Consider a discrete random variable X that takes values x_i in a vector space with probabilities \lambda_i \geq 0 where \sum \lambda_i = 1; then the expectation is given by E[X] = \sum \lambda_i x_i, which is precisely the convex combination of the x_i with weights \lambda_i.^[20] This representation extends to continuous cases via integration, where the expectation integrates the variable against a probability density, mirroring the convex structure.^[20] A key application is in mixture distributions, where a new probability measure is formed as a convex combination of component distributions. For instance, suppose we mix two Bernoulli distributions: one with success probability 0.2 and another with 0.8, using weights 0.6 and 0.4, respectively. The resulting mixture has probability mass function P(Y=1) = 0.6 \cdot 0.2 + 0.4 \cdot 0.8 = 0.44 and P(Y=0) = 1 - 0.44 = 0.56, yielding a Bernoulli distribution with success probability 0.44.^[21] Such mixtures model heterogeneous populations, like subpopulations with differing behaviors, and the convex weights represent the proportions of each component.^[22] Under mixtures, certain functions of probability measures inherit convexity properties. The moment-generating function (MGF) of a mixture distribution is the convex combination of the MGFs of the components: if M_j(t) is the MGF of the j-th component, then the mixture MGF is M(t) = \sum \lambda_j M_j(t).^[23] Similarly, the cumulative distribution function (CDF) of the mixture is F(y) = \sum \lambda_j F_j(y), a convex combination of the component CDFs F_j, preserving the non-decreasing and right-continuous nature of CDFs.^[21] Convex combinations also underpin the law of total expectation, which expresses the overall expectation as a mixture over conditional expectations. Specifically, E[X] = E[E[X \mid Y]] = \int E[X \mid Y = y] \, dF_Y(y), where the integral is a continuous convex combination weighted by the distribution of Y, or in discrete cases, E[X] = \sum P(Y = y_k) E[X \mid Y = y_k].^[20] This iterated form highlights how conditional mixtures aggregate to the unconditional expectation via convex weighting.^[20]

Applications

Convex Sets and Hulls

A convex set in a vector space is defined as a subset that contains all convex combinations of its elements. Specifically, for any points x_1, \dots, x_n in the set and nonnegative weights \lambda_1, \dots, \lambda_n summing to 1, the combination \sum_{i=1}^n \lambda_i x_i must also belong to the set.^[24] This closure property under finite convex combinations characterizes convexity equivalently to the line segment condition between any two points.^[25] The convex hull of a set S, denoted \operatorname{conv}(S), is the smallest convex set containing S. It consists precisely of all finite convex combinations of points from S.^[24] In finite-dimensional spaces, this hull can be generated by iteratively taking convex combinations of points in S until closure under the operation is achieved. For compact convex sets in locally convex Hausdorff topological vector spaces, the Kreĭn–Milman theorem states that the set equals the closed convex hull of its extreme points—points that cannot be expressed as nontrivial convex combinations of other points in the set.^[26] A representative example is the convex hull of three non-collinear points A, B, and C in the plane, which forms the filled triangle including all points inside and on the boundary. Any point within this triangle can be written as a convex combination \lambda A + \mu B + \nu C where \lambda + \mu + \nu = 1 and \lambda, \mu, \nu \geq 0. To compute this hull for a finite set of points, one approach involves identifying the extreme points via linear programming to check non-representability as combinations, then forming the set of all convex combinations among them, though efficient algorithms like the gift wrapping method approximate the boundary facets.^[24] For infinite sets, convex hulls remain defined via finite convex combinations.

Optimization

In convex optimization, the feasible set is defined as a convex set, meaning any point within it can be expressed as a convex combination of other points in the set. This property ensures that local optima coincide with global optima when the objective function is convex, facilitating efficient algorithmic solutions. A key tool leveraging convex combinations is Jensen's inequality, which states that for a convex function f and weights \lambda_i \geq 0 summing to 1, f\left(\sum_i \lambda_i x_i\right) \leq \sum_i \lambda_i f(x_i). This inequality bounds the value of the objective function at a convex combination of points by the weighted average of the function values, providing a foundational bound in optimization analyses and proving convergence in methods like gradient descent on convex problems. In linear programming, the simplex method exploits the structure of the feasible set as the convex hull of its vertices, known as basic feasible solutions (BFS).^[27] Each BFS corresponds to a vertex of the polytope, and any feasible solution is a convex combination of these vertices; the algorithm iteratively pivots between adjacent BFS to reach an optimal vertex solution, ensuring polynomial-time performance in practice for many instances.^[28] In economics, Nash equilibria in games with convex strategy sets often involve mixed strategies, which are convex combinations of pure strategies.^[29] For example, in a two-player zero-sum game like matching pennies, the unique Nash equilibrium requires each player to randomize equally over their two pure strategies (heads or tails), forming a 50-50 convex combination that makes the opponent indifferent.^[29] Convex combination schemes appear in numerical methods for approximation theory, particularly in finite element methods (FEM) for solving partial differential equations (PDEs). In FEM, barycentric coordinates express points within elements as convex combinations of vertex nodes, enabling the construction of shape functions that ensure linear completeness and partition of unity for accurate approximations on complex domains.^[30] This approach preserves convexity in the approximation space, aiding stability and error control in simulations of elliptic or hyperbolic PDEs.^[31]

Affine Combinations

An affine combination of points x_1, \dots, x_n in a real vector space is defined as a linear combination \sum_{i=1}^n \lambda_i x_i, where the coefficients \lambda_i \in \mathbb{R} satisfy the condition \sum_{i=1}^n \lambda_i = 1.^[32] This formulation generalizes the notion of a weighted average without restricting the weights to non-negative values, distinguishing it from convex combinations, which impose the additional requirement that each \lambda_i \geq 0.^[33] The formula for an affine combination can be expressed as:

\mathbf{y} = \sum_{i=1}^n \lambda_i \mathbf{x}_i, \quad \sum_{i=1}^n \lambda_i = 1, \quad \lambda_i \in \mathbb{R}.

^[34] Every convex combination qualifies as an affine combination, since the non-negativity constraint is a subset of the real-valued coefficients summing to unity, but the reverse is not true: affine combinations can yield points beyond the boundaries defined by convexity.^[33] The collection of all possible affine combinations from a given set of points forms the affine hull, defined as the smallest affine subspace containing those points.^[35] For example, in \mathbb{R}^2, the points (0,0) and (1,0) have the affine combination $2 \cdot (1,0) + (-1) \cdot (0,0) = (2,0), where the coefficients sum to 1 but include a negative value, positioning (2,0) outside the line segment (their convex hull) through extrapolation.^[34]

Barycentric Coordinates

Barycentric coordinates provide a way to express any point inside a simplex as a unique convex combination of its vertices. For a point P in an n-dimensional simplex with vertices V_0, V_1, \dots, V_n, the barycentric coordinates (\lambda_0, \lambda_1, \dots, \lambda_n) satisfy P = \sum_{i=0}^n \lambda_i V_i, where \sum_{i=0}^n \lambda_i = 1 and \lambda_i \geq 0 for all i. These coordinates are unique because the vertices of a simplex form an affinely independent set, ensuring a one-to-one correspondence between points in the simplex and such coefficient tuples.^[36]^[37] To compute the barycentric coordinates, solve the affine system P = \sum_{i=0}^n \lambda_i V_i subject to \sum_{i=0}^n \lambda_i = 1 and the non-negativity constraints \lambda_i \geq 0. This can be done by considering the position vectors relative to one vertex or using matrix methods, such as forming the matrix with columns (V_i - V_0, 1) for i = 1 to n and solving for the coordinates via Cramer's rule or inversion. The non-negativity ensures the point lies within the convex hull of the simplex.^[38]^[31] A concrete example occurs in a 2D triangle with vertices A, B, C and an interior point P. The barycentric coordinates (\lambda_A, \lambda_B, \lambda_C) are given by the ratios of the areas of the sub-triangles formed by P and the opposite edges to the total area of \triangle ABC:

\lambda_A = \frac{\text{area}(\triangle PBC)}{\text{area}(\triangle ABC)}, \quad \lambda_B = \frac{\text{area}(\triangle PCA)}{\text{area}(\triangle ABC)}, \quad \lambda_C = \frac{\text{area}(\triangle PAB)}{\text{area}(\triangle ABC)}.

These areas can be computed using the shoelace formula or vector cross products, and the coordinates sum to 1 while being non-negative for points inside the triangle. For instance, the centroid has coordinates (1/3, 1/3, 1/3).^[39]^[40] This concept generalizes to higher dimensions, where barycentric coordinates parameterize points in n-simplices using volumes of sub-simplices instead of areas. For arbitrary polytopes, extensions of barycentric coordinates allow representation as convex combinations of vertices, though uniqueness may not hold outside simplices. In computer graphics, these coordinates enable smooth interpolation of attributes like colors or textures across polygonal faces by weighting vertex values according to the \lambda_i, facilitating efficient rendering and deformation.^[31]^[41]

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