Convex combination

Builds on Vector space Affine combination

Definition

Let $V$ be a real vector space and let $x_1, \dots, x_k$ be points of $V$ , with $k \ge 1$ . A convex combination of $x_1, \dots, x_k$ is a point of the form $\lambda_1 x_1 + \cdots + \lambda_k x_k,$ where the real numbers $\lambda_1, \dots, \lambda_k$ satisfy $\lambda_i \ge 0$ for each $i$ and $\lambda_1 + \cdots + \lambda_k = 1$ .

Remarks

Example: in $\R^2$ , the point $(1, 1)$ is a convex combination of $(0, 0)$ , $(3, 0)$ and $(0, 3)$ , with weights $1/3$ , $1/3$ , $1/3$ . The weights are nonnegative and add up to $1$ , and $\tfrac13 (0,0) + \tfrac13 (3,0) + \tfrac13 (0,3) = (1, 1)$ .

Non-example: $2 \cdot (1, 0) + (-1) \cdot (0, 0) = (2, 0)$ uses weights that add up to $1$ , so it is an affine combination, but the weight $-1$ is negative, so it is not a convex combination.

A convex combination of two points $x, y$ is a point $(1 - t) x + t y$ with $0 \le t \le 1$ . The same definition works over any ordered field in place of $\R$ .

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Tag	`H56XFFM23H`
URI	`m2://H56XFFM23H`
Kind	definition
Writing	W0 stub
MSC	52A05, 52A20
Defines	Convex combination
Uses	Vector space, Affine combination

Convex combination

Definition

Remarks

Reading path

Further reading

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