Kirchberger-type theorems for separation by convex domains

Zsolt Lángi; Márton Naszódi

doi:10.1007/s10998-008-8185-6

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Periodica Mathematica Hungarica

Volume/Issue:: Volume 57: Issue 2

Kirchberger-type theorems for separation by convex domains

Authors:

Zsolt Lángi

Zsolt Lángi Budapest University of Technology Dept. of Geometry Egry József u. 1. Budapest Hungary 1111

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Márton Naszódi

Márton Naszódi University of Alberta Dept. of Math. and Stats. 632 Central Academic Building Edmonton AB Canada T6G 2G1

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Pages:: 185–196

Online Publication Date:: 10 Dec 2008

Publication Date:: 01 Dec 2008

Article Category:: Research Article

DOI:: https://doi.org/10.1007/s10998-008-8185-6

Keywords:: 52A35; 52A20; 52A99; separation; Theorem of Kirchberger; convex set

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Abstract

We say that a convex set K in ℝ^dstrictly 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.

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Periodica Mathematica Hungarica

Print ISSN:: 0031-5303

Online ISSN:: 1588-2829

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Periodica Mathematica Hungarica
Language	English
Size	B5
Year of Foundation	1971
Volumes per Year	2
Issues per Year	4
Founder	Bolyai János Matematikai Társulat - János Bolyai Mathematical Society
Founder's Address	H-1055 Budapest, Hungary Falk Miksa u. 12.I/4.
Publisher	Akadémiai Kiadó Springer Nature Switzerland AG
Publisher's Address	H-1117 Budapest, Hungary 1516 Budapest, PO Box 245. CH-6330 Cham, Switzerland Gewerbestrasse 11.
Responsible Publisher	Chief Executive Officer, Akadémiai Kiadó
ISSN	0031-5303 (Print)
ISSN	1588-2829 (Online)