A Helly-type theorem for the dimension of the kernel of a starshaped set
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- by Marilyn Breen PDF
- Proc. Amer. Math. Soc. 73 (1979), 233-236 Request permission
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: \[ \begin {array}{*{20}{c}} {f(2,1) = 4,} \hfill \\ {f(n,k) = f(n - 1,k) + n + 2\quad {\text {for}}\;3 \leqslant n\;{\text {and}}\;1 \leqslant k \leqslant n - 1,} \hfill \\ {f(n,n) = n + 1.} \hfill \\ \end {array} \]References
- Marilyn Breen, The dimension of the kernel of a planar set, Pacific J. Math. 82 (1979), no. 1, 15–21. MR 549829 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.
- 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
Additional Information
- © Copyright 1979 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 73 (1979), 233-236
- MSC: Primary 52A30; Secondary 52A35
- DOI: https://doi.org/10.1090/S0002-9939-1979-0516470-3
- MathSciNet review: 516470