On the monotonicity of the gradient of a convex function
George J. Minty · Pacific Journal of Mathematics · 1964
The object of this note is to present some elementary theorems concerning convex functions in ^-dimensions and, more generally, topological vector spaces.These theorems are all essentially generalizations of the theorem "the derivative of a convex function of one real variable is monotonic non-decreasing", and appear to have been overlooked in the literature.Let X be a topological vector space with real scalars, and Y the conjugate-space (space of continuous linear functionals) of X.We shall write y(x), for x e X, y e Y, as ζx, y) to facilitate applications to Hubert space.The convex (real-valued) function φ will always be presumed to have convex domain Da X, and satisfies the inequalityfor all x lf x 2 in D, all s ^ 0, t ^ 0, s + t = 1.The graph G of Φ is a subset of the topological vector space X + R, and it is obvious that the "set of points lying above the graph of Φ": A={(x, r): x e D, r^Φ(x)} is a convex set.(This condition is also sufficient for the convexity of φ.)DEFINITION 1. Asetίclx Fis called a monotonic set provided that, for all (x l9 y x ) and (x 2 , y 2 ) in E, ^ 0.DEFINITION 2. ([6]) For Del, a function F:D-+Y is called πionotonic provided the graph of F is a monotonic set.Now, it is well known that the conjugate space of X + B is Y + R, and that a closed hyperplane in X + R is of the form {(x, r): ζx, y o y + rr Q = a} for some y o e Y, r o e R, ae R. (See [2], p. 26, Theoreme 1.)This representation is non-unique, but if r 0 Φ 0, the equation ζx, y o y + rr Q = a can be solved for r, and the resulting equation is, in an obvious sense, unique.These facts motivate the following definition: DEFINITION 3. A gradient hyperplane iϊof Φ is a closed hyperplane of support to A, the set of points lying above the graph of Φ in X + R, such that Jϊcan be written in the form {(x, r): r = φ(x 0 ) + ζx -x 0 , y o y\.(Note the analogy with the first two terms of a Taylor-series for φ.) REMARK 1.This definition might be considered inappropriate if Φ is not everywhere-defined over X; this problem will not concern us here.