Convexity numbers of closed sets in $\mathbb R^n$

Authors:
Stefan Geschke and Menachem Kojman

Journal:
Proc. Amer. Math. Soc. **130** (2002), 2871-2881

MSC (1991):
Primary 05A20, 52A05, 03E17, 03E35; Secondary 03E75

DOI:
https://doi.org/10.1090/S0002-9939-02-06437-7

Published electronically:
March 25, 2002

MathSciNet review:
1908910

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: For $n>2$ let $\mathcal I_n$ be the $\sigma$-ideal in $\mathcal P(n^\omega )$ generated by all sets which do not contain $n$ equidistant points in the usual metric on $n^\omega$. For each $n>2$ a set $S_n$ is constructed in $\mathbb {R}^n$ so that the $\sigma$-ideal which is generated by the convex subsets of $S_n$ restricted to the convexity radical $K(S_n)$ is isomorphic to $\mathcal I_n$. Thus $\operatorname {cov}(\mathcal I_n)$ is equal to the least number of convex subsets required to cover $S_n$ — the *convexity number* of $S_n$. For every non-increasing function $f:\omega \setminus 2\to \{\kappa \in \operatorname {card}:\operatorname {cf}(\kappa )>\aleph _0\}$ we construct a model of set theory in which $\operatorname {cov}(\mathcal I_n)=f(n)$ for each $n\in \omega \setminus 2$. When $f$ is strictly decreasing up to $n$, $n-1$ uncountable cardinals are simultaneously realized as convexity numbers of closed subsets of $\mathbb {R}^n$. It is conjectured that $n$, but never more than $n$, different uncountable cardinals can occur simultaneously as convexity numbers of closed subsets of $\mathbb {R}^n$. This conjecture is true for $n=1$ and $n=2$.

- Tomek Bartoszyński and Haim Judah,
*Set theory*, A K Peters, Ltd., Wellesley, MA, 1995. On the structure of the real line. MR**1350295** - Andreas Blass,
*Simple cardinal characteristics of the continuum*, Set theory of the reals (Ramat Gan, 1991) Israel Math. Conf. Proc., vol. 6, Bar-Ilan Univ., Ramat Gan, 1993, pp. 63–90. MR**1234278** - S. Geschke, M. Kojman, W. Kubis, R. Schipperus,
*Convex decompositions in the plane and continuous pair colorings of the irrationals*, submitted. - Martin Goldstern and Saharon Shelah,
*Many simple cardinal invariants*, Arch. Math. Logic**32**(1993), no. 3, 203–221. MR**1201650**, DOI https://doi.org/10.1007/BF01375552 - T. Jech,
*Multiple forcing*, Cambridge Tracts in Mathematics, vol. 88, Cambridge University Press, Cambridge, 1986. MR**895139** - L. Newelski and A. Rosłanowski,
*The ideal determined by the unsymmetric game*, Proc. Amer. Math. Soc.**117**(1993), no. 3, 823–831. MR**1112500**, DOI https://doi.org/10.1090/S0002-9939-1993-1112500-6 - S. Shelah, J. Steprans,
*The covering numbers of Mycielski ideals are all equal*, J. Symbolic Logic**66**(2001), 707–718. - Juris Steprāns,
*Decomposing Euclidean space with a small number of smooth sets*, Trans. Amer. Math. Soc.**351**(1999), no. 4, 1461–1480. MR**1473455**, DOI https://doi.org/10.1090/S0002-9947-99-02197-2 - J. Zapletal,
*Isolating cardinal invariants*, preprint.

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (1991):
05A20,
52A05,
03E17,
03E35,
03E75

Retrieve articles in all journals with MSC (1991): 05A20, 52A05, 03E17, 03E35, 03E75

Additional Information

**Stefan Geschke**

Affiliation:
Freie Universität Berlin, Arnimallee 2-6, D-1495 Berlin

MR Author ID:
681801

Email:
geschke@math.fu-berlin.de

**Menachem Kojman**

Affiliation:
Department of Mathematics, Ben Gurion University of the Negev, Beer Sheva, Israel

Email:
kojman@math.bgu.ac.il

Keywords:
Convex cover,
convexity number,
$n$-space,
forcing extension,
covering number

Received by editor(s):
April 19, 2001

Received by editor(s) in revised form:
May 31, 2001

Published electronically:
March 25, 2002

Communicated by:
Carl G. Jockusch, Jr.

Article copyright:
© Copyright 2002
American Mathematical Society