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Convex transformations with Banach lattice range.

Roman Ger — 1987

Stochastica

A closed epigraph theorem for Jensen-convex mappings with values in Banach lattices with a strong unit is established. This allows one to reduce the examination of continuity of vector valued transformations to the case of convex real functionals. In particular, it is shown that a weakly continuous Jensen-convex mapping is continuous. A number of corollaries follow; among them, a characterization of continuous vector-valued convex transformations is given that answers a question raised by Ih-Ching...

An inconsistency equation involving means

Roman GerTomasz Kochanek — 2009

Colloquium Mathematicae

We show that any quasi-arithmetic mean A φ and any non-quasi-arithmetic mean M (reasonably regular) are inconsistent in the sense that the only solutions f of both equations f ( M ( x , y ) ) = A φ ( f ( x ) , f ( y ) ) and f ( A φ ( x , y ) ) = M ( f ( x ) , f ( y ) ) are the constant ones.

On some aspects of Jensen-Menger convexity.

Joanna GerRoman Ger — 1992

Stochastica

The paper contains various results concerning the so-called homogeneity sets for convex functions defined on convex subsets of some special metric spaces named G-space (cf. H. Busemann [1]). A closed graph theorem for such type mappings is also presented.

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