Classification of semispaces according to their types in infinite-dimensional vector spaces (Q1282524)

A convex subset \(C\) of a real vector space \(X\) is called a semispace, or a hemispace in the terminology of the reviewer and \textit{I. Singer} [Linear Algebra Appl. 110, 117-179 (1988; Zbl 0656.52004)], if its complement is also convex. The author establishes some properties of semispaces; in particular, he proves that the pair \((X\setminus C,C)\) is a cut of the space preordered by the relation \(\leq_C\) defined on \(X\) by \(x\leq_Cy \Leftrightarrow y-x\in O^+C\) (the relation cone of \(C)\). This means that, for any \(x\in X\setminus C\) and \(y\in C\), one has \(x\leq_Cy\) and not \(y\leq_Cx\). Four types of semispaces can, in principle, be distinguished according to whether \(X\setminus C\) or \(C\) have or have not a largest and a smallest element, respectively, but the one for which those largest and smallest element exist is proved to be impossible; on the contrary, any infinite-dimensional space (unlike finite-dimensional spaces) contains semispaces for which none of them exist (the other two types occur in any finite- or infinite-dimensional space).