A lemma on convex functionals in finite-dimensional spaces (Q1100719)

The authors prove the lemma below, then describe its applications to the subdifferentiability and cone radius of convex operators. (A convex operator is a mapping \(L: D\subset {\mathbb{R}}^ n\to {\mathbb{R}}^ n\) which is convex with respect to the ordering induced by a closed convex subcone of \(R^ n\) with vertex at 0.) Lemma: Let D be a convex subset of \({\mathbb{R}}^ n \)with nonempty interior, and let \(f_ 1,...,f_ s:D\to {\mathbb{R}}\) be convex functionals. Suppose that for some points \(x_ 0\in int D\) and \(x_ 1,...,x_ s\in D\), the map \(x\mapsto \sum^{s}_{i=1}f_ i(x)(x_ i-x_ 0)\) is constant on D. Then one has the inequality \(\sum^{s}_{i=1}f_ i(x_ i)\geq \sum^{s}_{i=1}f_ i(x_ 0)\).