Selfishness of convex bodies and discrete point sets

Let F be a family of sets in Rd. A set M⊂Rd is called F-convex if for any pair of distinct points x,y∈M, there is a set F∈F such that x,y∈F and F⊂M. A family F of compact sets is called complete if F contains all compact F-convex sets. Generalizing the definition in Yuan and Zamfirescu (2016), a com...

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Bibliographic Details
Published inEuropean journal of combinatorics Vol. 80; pp. 416 - 431
Main Authors Yuan, Liping, Zamfirescu, Tudor, Zhang, Yue
Format Journal Article
LanguageEnglish
Published Elsevier Ltd 01.08.2019
Online AccessGet full text
ISSN0195-6698
1095-9971
DOI10.1016/j.ejc.2018.02.012

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Summary:Let F be a family of sets in Rd. A set M⊂Rd is called F-convex if for any pair of distinct points x,y∈M, there is a set F∈F such that x,y∈F and F⊂M. A family F of compact sets is called complete if F contains all compact F-convex sets. Generalizing the definition in Yuan and Zamfirescu (2016), a compact set K will be called selfish, if the family FK of all sets similar to K contains all compact FK-convex sets. In this paper, we investigate the selfishness of rectangles, isosceles triangles, regular n-gons, and some finite sets.
ISSN:0195-6698
1095-9971
DOI:10.1016/j.ejc.2018.02.012