EDB — 184

Exercises

1. [184] Prerequisites:[181]↺↻,[17T]↺↻.Difficulty:*.Let \(C⊆ ℝ^ n\) be a convex set, let \(f:C→ℝ\) be a convex function, we fix \(z∈ {{C}^\circ }\): show that there exists \(v∈ℝ^ n\) such that

   \begin{equation} ∀ x∈ C, f(x)≥ f(z) + ⟨ v,x-z⟩~ ~ . \label{eq:piano_ sotto_ convessa} \end{equation}

   24

   The plane thus defined is called support plan for \(f\) in \(z\). Note:It is preferable not to assume that \(f\) is continuous, while proving this result, as this result is generally used to prove that \(f\) is continuous!.

   Solution 1

   [185]↺↻