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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

A Helly-type theorem for the dimension of the kernel of a starshaped set


Author: Marilyn Breen
Journal: Proc. Amer. Math. Soc. 73 (1979), 233-236
MSC: Primary 52A30; Secondary 52A35
MathSciNet review: 516470
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Abstract: This study will investigate the dimension of the kernel of a starshaped set, and the following result will be obtained: Let S be a compact set in some linear topological space L. For $ 1 \leqslant k \leqslant n$, the dimension of $ \ker S$ is at least k if and only if for some $ \varepsilon > 0$ and some n-dimensional flat $ {F^n}$ in L, every $ f(n,k)$ points of S see via S a common k-dimensional neighborhood in $ {F^n}$ having radius $ \varepsilon $. The number $ f(n,k)$ is defined inductively as follows:

\begin{displaymath}\begin{array}{*{20}{c}} {f(2,1) = 4,} \hfill \\ {f(n,k) = f(n... ...lant n - 1,} \hfill \\ {f(n,n) = n + 1.} \hfill \\ \end{array} \end{displaymath}


References [Enhancements On Off] (What's this?)

  • [1] Marilyn Breen, The dimension of the kernel of a planar set, Pacific J. Math. 82 (1979), no. 1, 15–21. MR 549829 (81h:52006)
  • [2] L. Danzer, B. Grünbaum and V. Klee, Helly's theorem and its relatives, Convexity, Proc. Sympos. Pure Math., vol. 7, Amer. Math. Soc., Providence, R. I., 1962, pp. 101-180.
  • [3] M. Krasnosselsky, Sur un critère pour qu’un domaine soit étoilé, Rec. Math. [Mat. Sbornik] N. S. 19(61) (1946), 309–310 (Russian, with French summary). MR 0020248 (8,525a)

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1979-0516470-3
PII: S 0002-9939(1979)0516470-3
Article copyright: © Copyright 1979 American Mathematical Society