Separating two compact sets by a parallelotope

Abstract:

Let P and Q be compact subsets of Euclidean n-space. Paul Kirchberger has established that there exists a closed half-space M such that $P \subset M$ and $Q \cap M = \emptyset$ iff for each set S consisting of $n + 2$ or fewer points of $P \cup Q$, there exists a closed half-space ${M_S}$ such that $(P \cap S) \subset {M_S}$ and $(Q \cap S) \cap {M_S} = \emptyset$. In this paper the problem of replacing the half-spaces by parallelotopes is considered, and the critical number of points in $P \cup Q$ is shown to be $n + 1$. Applications of this are drawn to systems of linear inequalities and to Carathéodory’s theorem.