# Definition:Barycenter

## Definition

Let $\mathcal E$ be an affine space over a field $k$.

Let $p_1,\ldots,p_n \in \mathcal E$ be points.

Let $\lambda_1,\ldots,\lambda_n \in k$ such that $\displaystyle \sum_{i \mathop = 1}^n \lambda_i = 1$.

The barycenter of $p_1,\ldots,p_n$ with weights $\lambda_1,\ldots,\lambda_n$ is the unique point $q$ of $\mathcal E$ such that for every point $r \in \mathcal E$

$\displaystyle q = r + \sum_{i \mathop = 1}^n\lambda_i \vec{r p_i}$

## Notation

It is conventional to write:

$q = \lambda_1 p_1 + \cdots + \lambda_np_n$

despite the fact that linear combinations are not defined in affine spaces.

## Also known as

UK orthography encodes this as barycentre.