Kirchberger-type theorems for separation by convex domains

Abstract

We say that a convex set K in ℝ^d strictly separates the set A from the set B if A ⊂ int(K) and B ⋂ cl K = ø. The well-known Theorem of Kirchberger states the following. If A and B are finite sets in ℝ^d with the property that for every T ⊂ A⋃B of cardinality at most d + 2, there is a half space strictly separating T ⋂ A and T ⋂ B, then there is a half space strictly separating A and B. In short, we say that the strict separation number of the family of half spaces in ℝ^d is d + 2.

In this note we investigate the problem of strict separation of two finite sets by the family of positive homothetic (resp., similar) copies of a closed, convex set. We prove Kirchberger-type theorems for the family of positive homothets of planar convex sets and for the family of homothets of certain polyhedral sets. Moreover, we provide examples that show that, for certain convex sets, the family of positive homothets (resp., the family of similar copies) has a large strict separation number, in some cases, infinity. Finally, we examine how our results translate to the setting of non-strict separation.