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

DOI:
https://doi.org/10.1090/S0002-9939-1979-0516470-3

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 , the dimension of is at least *k* if and only if for some and some *n*-dimensional flat in *L*, every points of *S* see via *S* a common *k*-dimensional neighborhood in having radius . The number is defined inductively as follows:

**[1]**Marilyn Breen,*The dimension of the kernel of a planar set*, Pacific J. Math.**82**(1979), no. 1, 15–21. MR**549829****[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**

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DOI:
https://doi.org/10.1090/S0002-9939-1979-0516470-3

Article copyright:
© Copyright 1979
American Mathematical Society