Convexity/What is a convex set?



A convex set is a set of points such that, given any two points A, B in that set, the line AB joining them lies entirely within that set.

Intuitively, this means that the set is connected (so that you can pass between any two points without leaving the set) and has no dents in its perimeter. Sections of the perimeter may be straight lines.

Theorem
If X is a convex set and x1 ... xk are any points in it, then

$$\displaystyle x = \sum_{i=1}^k \lambda_i x_i$$ where all $$\displaystyle \lambda_i > 0$$ and $$\displaystyle \sum_{i=1}^k \lambda_i=1$$

is also in X.

Proof: If $$k=2$$, the theorem is true by the definition of a convex set.

Now we proceed by induction. Assume the theorem is true for $$k=m$$. If $$k=m+1$$, we can formulate the definition of $$x$$ as a linear combination of two points, one of which is itself a linear combination of the first $$m$$ points and the other is the $$(m+1)$$th point:

$$\displaystyle x = L \sum_{i=1}^{m} (\lambda_i / L) \, x_i + (1 - L) \, x_{m + 1} $$ where $$\displaystyle L=\sum_{i=1}^m \lambda_i$$.

By the induction hypothesis, the first point must be in $$X$$; therefore, from the definition, $$x\in X$$.