The structure of hemispaces in \({\mathbb{R}}^ n\) (Q1110096)

A convex set C of Euclidean n-space \({\mathbb{R}}^ n\) such that \({\mathbb{R}}^ n\setminus C\) is also convex is called a hemispace of \({\mathbb{R}}^ n\). The authors of this extensive paper present several characterizations of hemispaces in terms of linear operators and lexicographical order. They also discuss affine transformations preserving a hemispace. The following separation property is shown: if the intersection of convex sets \(C_ 1\),..., \(C_ p\) is empty, then there exist hemispaces \(H_ 1\supset C_ 1\),..., \(H_ p\supset C_ p\) whose intersection is empty.