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  • 1 Budapest University of Technology Dept. of Geometry Egry József u. 1. Budapest Hungary 1111
  • 2 University of Alberta Dept. of Math. and Stats. 632 Central Academic Building Edmonton AB Canada T6G 2G1
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Abstract  

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 TAB of cardinality at most d + 2, there is a half space strictly separating TA and TB, 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.

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