Math Problem Solver

Let f : X → R be differentiable on the open convex set X ⊂ Rn . Prove:

a) f is (strictly) convex, if and only if

f (y) − f (x) ≥ (>) ∇f (x)>(y − x) ∀ x, y ∈ X.

b) Let f also be twice continuously differentiable. Prove

i) f convex ⇔ Hess(f (x)) 0 for all x ∈ X ,

ii) f strictly convex ⇐ Hess(f (x)) 0 for all x ∈ X ,

iii) f uniformly convex ⇔ d>Hess(f (x))d ≥ μ‖d‖2 for all x ∈ X , d ∈ Rn and

μ > 0 .

Why does ” ⇒ ” not hold in b) ii) ?