We say that a convex set K in ℝdstrictly 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 ℝd 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 ℝd 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.