# Convexity ranks in higher dimensions

Fundamenta Mathematicae (2000)

- Volume: 164, Issue: 2, page 143-163
- ISSN: 0016-2736

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topKojman, Menachem. "Convexity ranks in higher dimensions." Fundamenta Mathematicae 164.2 (2000): 143-163. <http://eudml.org/doc/212451>.

@article{Kojman2000,

abstract = {A subset of a vector space is called countably convex if it is a countable union of convex sets. Classification of countably convex subsets of topological vector spaces is addressed in this paper.
An ordinal-valued rank function ϱ is introduced to measure the complexity of local nonconvexity points in subsets of topological vector spaces. Then ϱ is used to give a necessary and sufficient condition for countable convexity of closed sets.
Theorem. Suppose that S is a closed subset of a Polish linear space. Then S is countably convex if and only if there exists $α < ω_1$ so that ϱ(x) < α for all x ∈ S.
Classification of countably convex closed subsets of Polish linear spaces follows then easily. A similar classification (by a different rank function) was previously known for closed subset of $ℝ^2$ [3].
As an application of ϱ to Banach space geometry, it is proved that for every $α < ω_1$, the unit sphere of C(ωα) with the sup-norm has rank α. Furthermore, a countable compact metric space K is determined by the rank of the unit sphere of C(K) with the natural sup-norm:
Theorem. If $K_1,K_1$ are countable compact metric spaces and $S_i$ is the unit sphere in $C(K_i)$ with the sup-norm, i = 1,2, then $ϱ(S_1) = ϱ(S_2)$ if and only if $K_1$ and $K_2$ are homeomorphic.
Uncountably convex closed sets are also studied in dimension n > 2 and are seen to be drastically more complicated than uncountably convex closed subsets of $ℝ^2$},

author = {Kojman, Menachem},

journal = {Fundamenta Mathematicae},

keywords = {convexity; convexity number; Polish vector space; continuum hypothesis; Cantor-Bendixson degree; convexity rank},

language = {eng},

number = {2},

pages = {143-163},

title = {Convexity ranks in higher dimensions},

url = {http://eudml.org/doc/212451},

volume = {164},

year = {2000},

}

TY - JOUR

AU - Kojman, Menachem

TI - Convexity ranks in higher dimensions

JO - Fundamenta Mathematicae

PY - 2000

VL - 164

IS - 2

SP - 143

EP - 163

AB - A subset of a vector space is called countably convex if it is a countable union of convex sets. Classification of countably convex subsets of topological vector spaces is addressed in this paper.
An ordinal-valued rank function ϱ is introduced to measure the complexity of local nonconvexity points in subsets of topological vector spaces. Then ϱ is used to give a necessary and sufficient condition for countable convexity of closed sets.
Theorem. Suppose that S is a closed subset of a Polish linear space. Then S is countably convex if and only if there exists $α < ω_1$ so that ϱ(x) < α for all x ∈ S.
Classification of countably convex closed subsets of Polish linear spaces follows then easily. A similar classification (by a different rank function) was previously known for closed subset of $ℝ^2$ [3].
As an application of ϱ to Banach space geometry, it is proved that for every $α < ω_1$, the unit sphere of C(ωα) with the sup-norm has rank α. Furthermore, a countable compact metric space K is determined by the rank of the unit sphere of C(K) with the natural sup-norm:
Theorem. If $K_1,K_1$ are countable compact metric spaces and $S_i$ is the unit sphere in $C(K_i)$ with the sup-norm, i = 1,2, then $ϱ(S_1) = ϱ(S_2)$ if and only if $K_1$ and $K_2$ are homeomorphic.
Uncountably convex closed sets are also studied in dimension n > 2 and are seen to be drastically more complicated than uncountably convex closed subsets of $ℝ^2$

LA - eng

KW - convexity; convexity number; Polish vector space; continuum hypothesis; Cantor-Bendixson degree; convexity rank

UR - http://eudml.org/doc/212451

ER -

## References

top- [1] M. Breen, A decomposition theorem for m-convex sets, Israel J. Math. 24 (1976), 211-216. Zbl0342.52005
- [2] M. Breen, An ${R}^{d}$ analogue of Valentine’s theorem on 3-convex sets, ibid., 206-210. Zbl0342.52007
- [3] M. Breen and D. C. Kay, General decomposition theorems for m-convex sets in the plane, ibid., 217-233. Zbl0342.52006
- [4] G. Cantor, Ueber unendliche, lineare Punktmannichfaltigkeiten, Nr. 6, Math. Ann. 23 (1884), 453-488.
- [5] M. M. Day, Strict convexity and smoothness, Trans. Amer. Math. Soc. 78 (1955), 516-528. Zbl0068.09101
- [6] H. G. Eggleston, A condition for a compact plane set to be a union of finitely many convex sets, Proc. Cambridge Philos. Soc. 76 (1974), 61-66. Zbl0282.52003
- [7] V. Fonf and M. Kojman, On countable convexity of ${G}_{\delta}$ sets, in preparation. Zbl0980.46007
- [8] D. C. Kay and M. D. Guay, Convexity and a certain property ${P}_{m}$, Israel J. Math. 8 (1970), 39-52. Zbl0203.24701
- [9] A. S. Kechris and A. Louveau, Descriptive Set Theory and the Structure of Sets of Uniqueness, London Math. Soc. Lecture Note Ser. 128, Cambridge Univ. Press, 1987. Zbl0642.42014
- [10] A. S. Kechris, A. Louveau and W. H. Woodin, The structure of σ-ideals of compact sets, Trans. Amer. Math. Soc. 301 (1987), 263-288. Zbl0633.03043
- [11] V. Klee, Dispersed Chebyshev sets and covering by balls, Math. Ann. 257 (1981), 251-260. Zbl0453.41021
- [12] M. Kojman, Cantor-Bendixson degrees and convexity in ${\mathbb{R}}^{2}$, Israel J. Math., in press.
- [13] M. Kojman, M. A. Perles and S. Shelah, Sets in a Euclidean space which are not a countable union of convex subsets, Israel J. Math. 70 (1990), 313-342. Zbl0742.52002
- [14] J. F. Lawrence, W. R. Hare, Jr. and J. W. Kenelly, Finite unions of convex sets, Proc. Amer. Math. Soc. 34 (1972), 225-228. Zbl0237.52001
- [15] A. J. Lazar and J. Lindenstrauss, Banach spaces whose duals are ${L}_{1}$ spaces and their representing matrices, Acta Math. 126 (1971), 165-194. Zbl0209.43201
- [16] J. Lindenstrauss and R. R. Phelps, Extreme point properties of convex bodies in reflexive Banach spaces, Israel J. Math. 6 (1968), 39-48. Zbl0157.43802
- [17] J. Matoušek and P. Valtr, On visibility and covering by convex sets, ibid. 113 (1999), 341-379. Zbl0958.52008
- [18] A. A. Milyutin, Ismorphisms of the spaces of continuous functions over compact sets of cardinality of the continuum, Teor. Funktsiĭ Funktsional. Anal. i Prilozhen. 2 (1966), 150-156.
- [19] M. A. Perles and S. Shelah, A closed n+1-convex set in ${R}^{2}$ is the union of ${n}^{6}$ convex sets, Israel J. Math. 70 (1990), 305-312.
- [10] F. A. Valentine, A three point convexity property, Pacific J. Math. 7 (1957), 1227-1235. Zbl0080.15401

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