Ger, Roman Convex transformations with Banach lattice range. (English) Zbl 0675.46012 Stochastica 11, No. 1, 13-23 (1987). A proof is given of a convex analogue of the Banach closed graph theorem, to the effect that if X is a real linear topological Baire space, Y is a Banach lattice with strong unit, D a non-empty open convex subset of X and f:D\(\to Y\) satisfies(i) f(\(\frac{x+y}{2})\leq \frac{f(x)+f(y)}{2}(x,y\in D)\) (ii) \(\{\) (x,y)\(\in D\times Y:f(x)\leq y\}\) is closed in \(D\times Y,\) then f is continuous. A number of corollaries is presented, related in particular to the work of Ih-Ching Hsu [Proc. Am. Math. Soc. 58, 119-123 (1976; Zbl 0334.46009)]. Reviewer: C.B.Huijsmans Cited in 3 Documents MSC: 46B42 Banach lattices 46A40 Ordered topological linear spaces, vector lattices 26A51 Convexity of real functions in one variable, generalizations Keywords:closed epigraph theorem; strong unit; Jensen-convex; convex analogue of the Banach closed graph theorem; real linear topological Baire space; Banach lattice with strong unit Citations:Zbl 0334.46009 PDFBibTeX XMLCite \textit{R. Ger}, Stochastica 11, No. 1, 13--23 (1987; Zbl 0675.46012) Full Text: EuDML