convex combination

Convex combination

A point $y$ is a convex combination of $x_1,\dots ,x_k$ if

$$y=\sum _{i=1}^k{\theta }_i{x}_i\text{where}\sum _{i=1}^k{\theta }_i=1,{\theta }_i\ge 0.$$

It is an affine combination with the extra constraint that the weights are nonnegative. Equivalently: a weighted average of the points, so $y$ always lands inside their convex hull (unlike affine/linear combinations where negative or large weights escape).

The two-point case $\theta x_1+(1-\theta )x_2,\theta \in [0,1]$ is a line segment.
A convex set is a set closed under convex combinations of its points.

Mixture / probability distribution

The weights ${\theta }_i\ge 0,\sum {\theta }_i=1$ are a probability distribution over the points. So a convex combination is the expectation $\mathbb{E}[x]$ when $x=x_i$ with probability ${\theta }_i$. This is the bridge to jensen’s inequality (and why convex/concave functions are defined via the two-point convex combination).