2. Convex combinations

Definition. A convex combination of points (or equivalently, vectors) $P_1,\dots, P_k$ is a linear combination $c_1 P_1 + \dots + c_k P_k$ in which

(i) the sum of the coefficients is $1$ and

(ii) the coefficients are nonnegative.

Equivalently, a convex combination is a weighted average in which the weights are nonnegative and add to $1$. The term convex combination comes from the connection with convexity shown in Theorems 3.1 and 3.2 below.

Examples. (1) ${\frac 12 } P_1 + {\frac 12} P_2$, the ordinary average of $P_1$ and $P_2$, is a convex combination of $P_1$ and $P_2$.

(2) More generally, the ordinary average of $k$ points $P_1$,...,$P_k$ is a convex combination of them.

(3) For two points $P$, $Q$, the points on the line segment $PQ$ have the form $P + t(P-Q)$ $=$ $(1-t)P + tQ$, where $0 \leq t \leq 1$, and so are convex combinations of $P$ and $Q$.

Theorem 3.1. If $S_0 = \{P_1,\dots, P_{k}\}$ (a finite subset of R$^n$), then the convex hull of $S_0$ consists of every point that is a convex combination of $P_1$,...,$P_k$.

Theorem 3.2. If $S_0$ is any subset of R$^n$, then the convex hull of $S_0$ consists of every point that is a convex combination of a finite subset of $S_0$.

Note. A linear combination in which the coefficients have sum 1 is called a barycentric combination. Thus a convex combination is a barycentric combination in which the coefficients are also nonnegative.