Convexity

Convex Combination

January 29, 2026
AI, ML
Mathematics, Linear Algebra, Convexity

Convex Combination of Two Points #

A convex combination describes how to form a point between two points using weighted averages.

It is a fundamental building block in several advanced fields:

  • Linear Algebra & Geometry
  • Optimization Theory
  • Machine Learning (Specifically in SVMs, clustering, and data interpolation)

Given two points (or vectors) $\mathbf{x}_1, \mathbf{x}_2 \in \mathbb{R}^n$, a convex combination of these points is defined as:

$$\mathbf{x} = \lambda \mathbf{x}_1 + (1 - \lambda)\mathbf{x}_2$$

Where: