Convexity ranks in higher dimensions

• Autorzy: Menachem Kojman
• Języki publikacji: EN

Abstrakty

EN

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 $ℝ^2$

Słowa kluczowe

EN

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

Kategorie tematyczne

• 03E05: Other combinatorial set theory
• 52A07: Convex sets in topological vector spaces

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Fundamenta Mathematicae
• Rocznik: 2000
• Tom: 164
• Numer: 2
• Strony: 143-163

Daty

wydano

2000

otrzymano

1999-06-07

poprawiono

2000-02-01

Twórcy

autor

Menachem Kojman

• Department of Mathematics and Computer Science, Ben Gurion University of the Negev, Beer Sheva, Israel

Bibliografia

• [1] M. Breen, A decomposition theorem for m-convex sets, Israel J. Math. 24 (1976), 211-216.
• [2] M. Breen, An $R^d$ analogue of Valentine's theorem on 3-convex sets, ibid., 206-210.
• [3] M. Breen and D. C. Kay, General decomposition theorems for m-convex sets in the plane, ibid., 217-233.
• [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.
• [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.
• [7] V. Fonf and M. Kojman, On countable convexity of $G_δ$ sets, in preparation.
• [8] D. C. Kay and M. D. Guay, Convexity and a certain property $P_m$, Israel J. Math. 8 (1970), 39-52.
• [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.
• [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.
• [11] V. Klee, Dispersed Chebyshev sets and covering by balls, Math. Ann. 257 (1981), 251-260.
• [12] M. Kojman, Cantor-Bendixson degrees and convexity in $ℝ^2$, 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.
• [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.
• [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.
• [16] J. Lindenstrauss and R. R. Phelps, Extreme point properties of convex bodies in reflexive Banach spaces, Israel J. Math. 6 (1968), 39-48.
• [17] J. Matoušek and P. Valtr, On visibility and covering by convex sets, ibid. 113 (1999), 341-379.
• [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.