Characterizations of the generalized convex kernel
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- by Arthur G. Sparks
- Proc. Amer. Math. Soc. 27 (1971), 563-565
- DOI: https://doi.org/10.1090/S0002-9939-1971-0279692-8
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Abstract:
It is well known that the convex kernel $K$ of a set $S$ is the intersection of all maximal convex subsets of $S$. In this paper it is shown that the $n$th order kernel of a compact, simply-connected set $S$ in the plane is an ${L_n}$ set and is, in fact, the intersection of all maximal ${L_n}$ subsets of $S$. Furthermore, it is shown that one does not have to intersect the family of all the maximal ${L_n}$ subsets to obtain the $n$th order kernel, but that any subfamily thereof which covers the set is sufficient.References
- A. M. Bruckner and J. B. Bruckner, Generalized convex kernels, Israel J. Math. 2 (1964), 27–32. MR 171217, DOI 10.1007/BF02759731
- Arthur G. Sparks, Intersections of maximal $L_{n}$ sets, Proc. Amer. Math. Soc. 24 (1970), 245–250. MR 253153, DOI 10.1090/S0002-9939-1970-0253153-3
- F. A. Toranzos, Radial functions of convex and star-shaped bodies, Amer. Math. Monthly 74 (1967), 278–280. MR 208469, DOI 10.2307/2316022
Bibliographic Information
- © Copyright 1971 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 27 (1971), 563-565
- MSC: Primary 52.30
- DOI: https://doi.org/10.1090/S0002-9939-1971-0279692-8
- MathSciNet review: 0279692