Paper Info
Title
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.
Year
DOI
Venue
2008
10.1007/s10998-008-8185-6
Periodica Mathematica Hungarica
Keywords
Field
DocType
separation,Theorem of Kirchberger,convex set,52A35,52A20,52A99
Combinatorics,Finite set,Mathematical analysis,Convex set,Cardinality,Regular polygon,Half-space,Mathematics
Journal
Volume
Issue
ISSN
57
2
1588-2829
Citations 
PageRank 
References 
0
0.34
2
Authors
2
Name
Order
Citations
PageRank
Zsolt LáNgi101.35
Marton Naszodi2217.87