Continuous affine support mappings for convex operators (Q1120809)

The author deals with the properties of a measurable convex operator with values in an ordered locally convex vector space. In Section 1, the author gives sufficient conditions for an operator \(\phi\) ensuring that every linear support mapping of \(\phi\) is continuous. In Section 2, the author generalizes the classical result stating that a real-valued convex (resp. sublinear mapping) is the envelope of its continuous affine (resp. linear support functionals) to convex operators (resp. sublinear operator). In Section 3, the author presents a version of the sandwich theorem analoguous to a result of \textit{J. Zowe} [J. Math. Anal. Appl. 66, 282-296 (1978; Zbl 0389.46003)]. In Section 4, the author proves the existence of an affine continuous support mapping (resp. continuous linear support mapping) for a convex operator (resp. sublinear operator) by using his sandwich theorem. A necessary and sufficient condition for the existence of a continuous affine support mapping h for a convex operator \(\phi\) such that \(h(x_ 0)=\phi (x_ 0)\) for a fixed point \(x_ 0\) is obtained.