A convex combination is a fundamental concept in mathematics and optimization. It represents a weighted average of points where all weights are non-negative and sum to one. For example, given two points A and B, any point on the line segment between them can be expressed as a convex combination of A and B.
Convex combinations have several important properties. The weights must be non-negative and sum to one. The resulting point always lies within the convex hull of the original points. Here we see a triangle formed by three points, and any convex combination of these points will lie inside or on the boundary of this triangle, demonstrating how convex combinations preserve the convex structure.
Geometrically, convex combinations have a clear interpretation. In two dimensions, they create line segments between points. In higher dimensions, they form the convex hull - the smallest convex set containing all the points. Any convex combination will always lie within this convex hull, making it a fundamental concept for understanding geometric shapes and optimization problems.
Convex combinations have numerous practical applications. In finance, portfolio theory uses convex combinations to blend different assets, where weights represent the proportion of investment in each asset. In computer graphics, they're used for color blending and animation interpolation. Machine learning uses them in mixture models and ensemble methods. The key insight is that convex combinations provide a mathematically sound way to create weighted averages that preserve important properties.