By H. H. Schaefer
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Extra resources for Banach Lattices and Positive Operators (Grundlehren Der Mathematischen Wissenschaften Series, Vol 215)
1. 35. Prove that a function of one variable is homogeneous of degree α if and only if it is of the form f (x) = kxα . 36. Let f be a linearly homogeneous function deﬁned on a convex cone C ⊆ n . Prove that f is concave if and only if f (x + y) ≥ f (x) + f (y) for every x, y ∈ C. 37. A relevant result regarding homogeneous functions is Euler’s Theorem: let f be a diﬀerentiable function deﬁned on the open convex cone C ⊆ n . Then, f is homogeneous of degree α if and only if xT ∇f (x) = αf (x), ∀x ∈ C.
9. Let f be a continuous diﬀerentiable function on an open convex set S ⊆ n . Then, f is pseudoconvex (strictly pseudoconvex) on S if and only if the following conditions hold: (i) f is quasiconvex on S; (ii) If x0 ∈ S, ∇f (x0 ) = 0, then x0 is a local minimum (strict local minimum) for f . Proof. 5. Assume now that (i) and (ii) hold. 8 we must prove that if x0 ∈ S and u ∈ n are such that uT ∇f (x0 ) = 0, the function ϕ(t) = f (x0 + tu) attains a local minimum at t = 0. If ∇f (x0 ) = 0, the thesis follows from (ii); if ∇f (x0 ) = 0, the continuity of the gradient map implies the existence of an open neighbourhood U (x0 ) of x0 such that ∇f (x0 ) = 0, ∀x ∈ U (x0 ).
Consider the restriction ϕ(t) = f (x1 + t(x2 − x1 )), t ∈ [0, 1]. 2) holds. 2) holds. 1, there exist t1 , t2 ∈ [0, 1] such that ϕ(t1 ) ≥ ϕ(t2 ) and ϕ (t1 )(t2 − t1 ) > 0. Set x ¯1 = x1 + t1 (x2 − x1 ), x ¯2 = x1 + t2 (x2 − x1 ); we have f (¯ x1 ) ≥ f (¯ x2 ) 1 T and ϕ (t1 ) = (x2 − x1 )T ∇f (¯ x1 ) = t2 −t (¯ x − x ¯ ) ∇f (¯ x ). 2). 2) we also have a strict inequality on the right-hand side when S is open and x1 is not a critical point. 1 The diﬀerentiability of a function g on a set X ⊆ on an open set containing X.