Convex hulls and convex combinations

Question

Why is it possible to represent every point contained in a convex hull as a convex combination of the points that generate the convex hull?

I am studying convex hulls for linear programming. I have not understood the above. I have understood that if I have only $2$ points (it's a line beetween them), but with more points I don't get the idea. Can you show me?

Answer 1

Any convex combination of three or more points can be written as a "chain" of convex combinations of two points. For example, $\frac13 A + \frac13B + \frac13C$ can be expressed as $\frac13 A + \frac23(\frac12B + \frac12C)$: first we take the convex combination $\frac12B + \frac12C$, call the result $X$, and then take the convex combination $\frac13A + \frac23X$.

So if you understand why the convex combinations of two points form a line segment, you should understand why the convex combinations of three points form a triangle. If the points are $A$, $B$, and $C$, then:

• We can get every point $X$ on line segment $BC$ as a convex combination of $A$, $B$, and $C$.
• So any point on line segment $AX$ is a convex combination of $A$ and $X$: this, in turn is a convex combination of $A$, $B$, and $C$.
• As we move $X$ from $B$ to $C$, the segments $AX$ sweep out the entire area of $\triangle ABC$, as shown in the diagram below.