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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Convexity numbers of closed sets in $\mathbb R^n$
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by Stefan Geschke and Menachem Kojman PDF
Proc. Amer. Math. Soc. 130 (2002), 2871-2881 Request permission

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 $n=2$.
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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
  • 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.
  • © Copyright 2002 American Mathematical Society
  • 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
  • MathSciNet review: 1908910