Convex combination

A weighted sum of points with nonnegative weights that add to one. It lies in the convex hull of the points being combined.

Convex set: A set containing the line segment between every pair of its points. Convex sets contain every convex combination of their points.
Simplex: The higher-dimensional generalization of a triangle or tetrahedron. Its points correspond to convex combinations of its vertices.
Mixture model: A probabilistic model whose distribution is a weighted sum of component distributions. Its mixing proportions are nonnegative weights summing to one.
Linear combination: A sum of vectors multiplied by arbitrary scalar coefficients. Unlike a convex combination, its coefficients need not be nonnegative or sum to one.
Affine combination: A weighted sum of points whose weights add to one, without requiring nonnegative weights. Dropping nonnegativity gives affine combinations, which can lie outside the points’ convex hull.
Carathéodory's theorem: A theorem bounding how many points are needed to represent a point in a Euclidean convex hull. It limits the number of nonzero weights needed in a convex combination.
Expected value: The probability-weighted average of the possible values of a random variable. For a finite distribution, the expectation is a convex combination of its outcomes.
Conical combination: A weighted sum with nonnegative coefficients, without requiring them to sum to one. It keeps nonnegative weights but removes the normalization condition.
Barycentric coordinates: Coordinates that express a point as an affine combination of vertices. Nonnegative barycentric coordinates characterize points inside a simplex.
Extreme point: A point in a convex set that lies in no open segment contained in that set. Extreme points cannot be written as nontrivial convex combinations of other points in the set.

Linked from 12 pages