EDB — 180

Exercises

1. [180]Let \(C⊂ ℝ^ n\) be a convex set. Let \(f : C →ℝ\) be convex; let \(x_ 1,\ldots ,x_ n ∈ C\) and \(t_ 1,\ldots ,t_ n ∈ [0, 1]\) be such that \(∑_{i=1}^ n t_ i = 1\). Show that

   \[ ∑_{i=1}^ n t_ i x_ i∈ C \]

   and

   \[ f \left(∑_{i=1}^ n t_ i x_ i\right)≤ ∑_{i=1}^ n t_ i f (x_ i ) ~ . \]