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Qué (quién) es convex combination - definición

LINEAR COMBINATION OF POINTS WHOSE COEFFICIENTS ARE NON-NEGATIVE AND SUM TO ONE

Convex sum

$Convex combination of three points <math>v_{0},v_{1},v_{2} \text{ of } 2\text{-simplex} \in \mathbb{R}^{2}</math> in a two dimensional vector space <math> \mathbb{R}^{2}</math> as shown in animation with <math>\alpha^{0}+\alpha^{1}+\alpha^{2}=1</math>, <math>P( \alpha^{0},\alpha^{1},\alpha^{2} )</math> <math>:= \alpha^{0} v_{0} + \alpha^{1} v_{1} + \alpha^{2} v_{2}</math> . When P is inside of the triangle <math>\alpha_{i}\ge 0</math>. Otherwise, when P is outside of the triangle, at least one of the <math>\alpha_{i}</math> is negative.$
$Convex combination of four points <math>A_{1},A_{2},A_{3},A_{4} \in \mathbb{R}^{3}</math> in a three dimensional vector space <math> \mathbb{R}^{3}</math> as animation in [[Geogebra]] with <math>\sum_{i=1}^{4} \alpha_{i}=1</math> and <small><math>\sum_{i=1}^{4} \alpha_{i}\cdot A_{i}=P</math></small>. When P is inside of the tetrahedron <math>\alpha_{i}>=0</math>. Otherwise, when P is outside of the tetrahedron, at least one of the <math>\alpha_{i}</math> is negative.$
$Convex combination of two functions as vectors in a vector space of functions - visualized in Open Source Geogebra with <math>[a,b]=[-4,7]</math> and as the first function <math>f:[a,b]\to \mathbb{R}</math> a polynomial is defined. <math>f(x):= \frac{3}{10} \cdot x^2 - 2 </math> A trigonometric function <math>g:[a,b]\to \mathbb{R}</math> was chosen as the second function. <math>g(x):= 2 \cdot cos(x) + 1</math> The figure illustrates the convex combination <math>K(t):= (1-t)\cdot f + t \cdot g</math> of <math>f</math> and <math>g</math> as graph in red color.$
$Convex combination of two points <math> v_1,v_2 \in \mathbb{R}^2</math> in a two dimensional vector space <math>\mathbb{R}^2</math> as animation in [[Geogebra]] with <math>t \in [0,1]</math> and <math> K(t) := (1-t)\cdot v_1 + t \cdot v_2</math>$

convex hull

NOTION IN TOPOLOGICAL VECTOR SPACES

Convex envelope; Closed convex hull; Convex Hull; Convex span; Convex closure; Minimum convex polygon; Applications of convex hulls

<mathematics, graphics> For a set S in space, the smallest convex set containing S. In the plane, the convex hull can be visualized as the shape assumed by a rubber band that has been stretched around the set S and released to conform as closely as possible to S. (1997-08-03)

Convex hull

NOTION IN TOPOLOGICAL VECTOR SPACES

Convex envelope; Closed convex hull; Convex Hull; Convex span; Convex closure; Minimum convex polygon; Applications of convex hulls

In geometry, the convex hull or 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.

https://en.wikipedia.org/wiki/Convex_hull

Locally convex topological vector space

TYPE OF TOPOLOGICAL VECTOR SPACE

Locally convex; Locally convex space; Locally convex spaces; Locally convex topology; Locally convex basis; Locally convex vector space; LCTVS; Finest locally convex topology

In functional analysis and related areas of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topological vector spaces whose topology is generated by translations of balanced, absorbent, convex sets.

https://en.wikipedia.org/wiki/Locally_convex_topological_vector_space

Wikipedia

Convex combination

In convex geometry and vector algebra, a convex combination is a linear combination of points (which can be vectors, scalars, or more generally points in an affine space) where all coefficients are non-negative and sum to 1. In other words, the operation is equivalent to a standard weighted average, but whose weights are expressed as a percent of the total weight, instead of as a fraction of the count of the weights as in a standard weighted average.

More formally, given a finite number of points $x_{1},x_{2},\dots ,x_{n}$ in a real vector space, a convex combination of these points is a point of the form

\alpha _{1}x_{1}+\alpha _{2}x_{2}+\cdots +\alpha _{n}x_{n}

where the real numbers $\alpha _{i}$ satisfy $\alpha _{i}\geq 0$ and $\alpha _{1}+\alpha _{2}+\cdots +\alpha _{n}=1.$

As a particular example, every convex combination of two points lies on the line segment between the points.

A set is convex if it contains all convex combinations of its points. The convex hull of a given set of points is identical to the set of all their convex combinations.

There exist subsets of a vector space that are not closed under linear combinations but are closed under convex combinations. For example, the interval $[0,1]$ is convex but generates the real-number line under linear combinations. Another example is the convex set of probability distributions, as linear combinations preserve neither nonnegativity nor affinity (i.e., having total integral one).

https://en.wikipedia.org/wiki/Convex combination