How would one show that the convex combinations of points from a convex set are also in that convex set.

Question

I am aware of the theorem "a set C is convex iff any convex combination of points in C is in C" but I was wondering how to prove one side of this theorem, if a set C is convex how would you show that the convex combination of any points in C is also in C? I am aware of the definitions of a convex set and a convex combination but I can't figure out how to prove this part, am I missing a property of convex sets that makes this proof more straightforward?

Answer 1

Use mathematical induction. It is true for two points from the definition of convexity. Suppose it is true for $k-1$ points.

If $\sum_{i=1}^k \lambda_i=1$, if $\lambda_k \ne 1$, and $x_1, \ldots, x_k \in C$

\begin{align}\sum_{i=1}^k \lambda_i x_i &= \sum_{i=1}^{k-1} \lambda_i x_i + \lambda_kx_k \\ &= (1-\lambda_k)\left(\sum_{i=1}^{k-1}\frac{\lambda_i}{1-\lambda_k}x_i\right) + \lambda_kx_k\\\end{align}

Notice that the $\sum_{i=1}^{k-1}\frac{\lambda_i}{1-\lambda_k}$ would sum to $1$.