On the lower hull of convex functions (Q1262400)

In generalization of a result of \textit{F. Bernstein} and \textit{G. Doetsch} [Math. Ann. 76, 514-526 (1915)] the authors offer the following as main result. Let X be a real linear Baire space, D(\(\subset X)\) convex and open, and f:D\(\to [-\infty,\infty [\) midpoint-convex. Then \[ m_ f(x)=\sup_{U\in {\mathcal F}_ x}\inf_{t\in U\cap D}f(t) \] is continuous and convex. Here \({\mathcal F}_ x=\{U| x\in U\subset D\), U open\(\}\). Several related questions are discussed and a rich array of results is presented.