By Weil W.

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Hence, f is convex, if and only if f (αx1 + (1 − α)x2 ) ≤ α1 β1 + (1 − α)β2 , for all x1 , x2 ∈ Rn , α ∈ [0, 1] and all β1 ≥ f (x1 ), β2 ≥ f (x2 ). Then, it is necessary and sufficient that this inequality is satisfied for β1 = f (x1 ), β2 = f (x2 ), and we obtain the assertion. Remarks. (1) A function f : Rn → R is affine, if and only if f is convex and concave. If f is affine, then epi f is a half-space in Rn+1 (and dom f = Rn ). (2) For a convex function f , the sublevel sets {f < α} and {f ≤ α} are convex.

Thus, for a convex function f we exclude the value −∞, whereas for a concave function we exclude ∞. (2) If A ⊂ Rn is a subset, a function f : A → (−∞, ∞) is called convex, if the extended function f˜ : Rn → (−∞, ∞], given by f˜ := f ∞ on A, Rn \ A , is convex. This automatically requires that A is a convex set. In view of this construction, we need not consider convex functions defined on subsets of Rn , but we rather can assume that convex functions are always defined on all of Rn . 39 40 CHAPTER 2.

In view of this construction, we need not consider convex functions defined on subsets of Rn , but we rather can assume that convex functions are always defined on all of Rn . 39 40 CHAPTER 2. CONVEX FUNCTIONS (3) On the other hand, we often are only interested in convex functions f : Rn → (−∞, ∞] at points, where f is finite. We call dom f := {x ∈ Rn : f (x) < ∞} the effective domain of the function f : Rn → (−∞, ∞]. For a convex function f , the effective domain dom f is convex. (4) The function f ≡ ∞ is convex, it is called the improper convex function; convex functions f with f ≡ ∞ are called proper.