# Convexity ranks in higher dimensions

Fundamenta Mathematicae (2000)

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

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## Abstract

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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 $\alpha <{\omega }_{1}$ 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 $\alpha <{\omega }_{1}$, 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 $C\left({K}_{i}\right)$ with the sup-norm, i = 1,2, then $\varrho \left({S}_{1}\right)=\varrho \left({S}_{2}\right)$ 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 ${ℝ}^{2}$

## How to cite

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Kojman, 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

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