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Starshapedness in convexity spaces. (English) Zbl 0584.52001
Let (X,\({\mathcal C})\) be a convexity space. For \(S\subset X\), let \({\mathcal C}- \ker (S)=\{p\in S:{\mathcal C}(p,x)\subset S\quad for\quad all\quad x\in S\}\) be the kernel of S. The space (X,\({\mathcal C})\) is said to be a B-space (T- space) iff for each \(S\subset X\) \({\mathcal C}-\ker (S)\) is convex (resp., is the intersection of all maximal convex subsets of S). A B-space is always domain finite, and a domain-finite space is a B-space only if it is join- hull commutative.
The two main results are: 1) In a B-space, S is convex iff \({\mathcal C}\)- ker(S)\(=S\), 2) If all singletons are convex, then (X,\({\mathcal C})\) is a T- space iff it is domain finite and join-hull commutative.
Using the first result the solution of the linearization problem is given for a larger class of convexity spaces than in the paper of P. Mah, S. A. Naimpally, and J. H. M. Whitfield in J. Lond. Math. Soc., II. Ser. 13, 209-214 (1976; Zbl 0326.52005].
Reviewer: J.Cirulis

MSC:
52A01 Axiomatic and generalized convexity
52A30 Variants of convex sets (star-shaped, (\(m, n\))-convex, etc.)
52A05 Convex sets without dimension restrictions (aspects of convex geometry)
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References:
[1] H. Brunn : Über Kemeigebiete . Math. Ann. 73 (1913) 436-440. · JFM 44.0560.01
[2] V.W. Bryant and R.J. Webster : Convexity spaces. I. The basic properties . J. Math. Anal. Appl. 37 (1972) 206-213. · Zbl 0197.48401 · doi:10.1016/0022-247X(72)90268-5
[3] E. Degreef : Some results in generalized convexity . Doct. diss., Free University of Brussel (1981) · Zbl 0436.52003
[4] P.C. Hammer : Extended topology: Domain finiteness . Indag. Math. 25 (1963) 200-212. · Zbl 0118.17702
[5] R.E. Jamison : A general theory of convexity . Doct. diss., University of Washington, Seattle (1974).
[6] D.C. Kay and E.W. Womble : Axiomatic convexity theory and relationships between the Carathéodory, Helly and Radon numbers . Pacific J. Math. 38 (1971) 471-485. · Zbl 0235.52001 · doi:10.2140/pjm.1971.38.471
[7] K. Kołodziejczyk : On starshapedness of the union of closedsets in Rn . Colloq. Math. (to appear). · Zbl 0635.52006
[8] K. Kołodziejczyk : The starshapedness number of a convexity space (preprint). · Zbl 0584.52001 · numdam:CM_1985__56_3_361_0 · eudml:89746
[9] M.A. Krasnosel’Skii : Sur un critère pour qu’un domain soit étoilé . Mat. Sb. 19 (1946) 309-310. · Zbl 0061.37705
[10] F.W. Levi : On Helly’s theorem and the axioms of convexity . J. Indian Math. Soc. 15 (1951) 65-76. · Zbl 0044.19101
[11] P. Mah , S.A. Naimpally and J.H.M. Whitfield : Linearization of a convexity space . J. London Math. Soc. 13 (1976) 209-214. · Zbl 0326.52005 · doi:10.1112/jlms/s2-13.2.209
[12] G. Sierksma : Axiomatic convexity theory and the convex product space . Doct. diss., University of Groningen (1976). · Zbl 0336.52001
[13] V.P. Soltan : Starshaped sets in the axiomatic theory of convexity . Bull. Acad. Sci. Georgian SSR 96 (1979) 45-48. · Zbl 0418.52003
[14] F.A. Toranzos : Radial functions of convex and starshaped sets . Amer. Math. Monthly 74 (1967) 278-280. · Zbl 0145.42802 · doi:10.2307/2316022
[15] F.A. Valentine : Convex Sets . New York: McGraw-Hill (1964). · Zbl 0129.37203
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