EDB — 182

[182]Let \(C⊆ ℝ^ n\) be a convex set, and \(f:C→ℝ\) a convex function. Given \(l\in {\mathbb {R}}\), define the sublevel set as

Show that \(L_ l\) is a convex (possibly empty) set. Deduce that the minimum points of \(f\) are a convex (possibly empty) set. Show that if \(f\) is strictly convex there can be at most one minimum point.