## Franklin's Notes

### Affine combination

In an affine space , there is no natural way of extending the operation of vector addition, since vector addition cannot be defined independently of a choice of origin (nor can scalar multiplication). However, it is possible to define weighted averages or "centroids" of collections of points. Given $x,y\in X$, we defined the affine combination $\mu x+\lambda y$ to equal the point $x+\lambda d(x,y)$ or equivalently the point $y+\mu d(y,x)$, so long as $\lambda,\mu$ satisfy $\lambda+\mu=1$. When $\lambda$ and $\mu$ are both positive, $\mu x + \lambda y$ is "between" $x$ and $y$, or on the line segment joining them.

Why is this definition unambiguous when $\lambda+\mu=1$? We can prove this using the identity for the inverses of the partially-applied distance maps, through the following sequence of equalities:

We can similarly define affine combinations of several points with the modified constraint $\lambda_1+\lambda_2+\cdots + \lambda_n=1$.