EDB — 16Y

Exercises

1. [16Y] Let \(C⊆ ℝ^ n\) be a set; show that it is convex if and only if it contains every convex combination of its points, that is: for every \(k≥ 1\), for every choice of \(x_ 1,\ldots x_ k∈ C\) , for each choice \(t_ 1,\ldots t_ k≥ 0\) with \(t_ 1+\cdots + t_ k=1\), you have

   \[ x_ 1 t_ 1+\cdots + x_ k t_ k ∈ C\quad . \]