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

Admissible kernels for starshaped sets


Author: Marilyn Breen
Journal: Proc. Amer. Math. Soc. 82 (1981), 622-628
MSC: Primary 52A30
MathSciNet review: 614890
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Abstract: Steven Lay has posed the following interesting question: If $ D$ is a convex subset of $ {{\mathbf{R}}^d}$, then is there a starshaped set $ S \ne D$ in $ {{\mathbf{R}}^d}$ whose kernel is $ D$? Thus the problem is that of characterizing those convex sets which are admissible as the kernel of some nonconvex starshaped set in $ {{\mathbf{R}}^d}$. Here Lay's problem is investigated for closed sets, and the following results are obtained: If $ D$ is a nonempty closed convex subset of $ {{\mathbf{R}}^2}$, then $ D$ is the kernel of some planar set $ S \ne D$ if and only if $ D$ contains no line. If $ D$ is a compact convex set in $ {{\mathbf{R}}^d}$, then there is a compact set $ S \ne D$ in $ {{\mathbf{R}}^d}$ whose kernel is $ D$.


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

  • [1] Steven Lay, Proceedings of Conference on Convexity and Related Combinatorics, Dekker, New York, (to appear).
  • [2] Frederick A. Valentine, Convex sets, McGraw-Hill Series in Higher Mathematics, McGraw-Hill Book Co., New York, 1964. MR 0170264 (30 #503)

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1981-0614890-9
PII: S 0002-9939(1981)0614890-9
Keywords: Starshaped sets, convex kernel
Article copyright: © Copyright 1981 American Mathematical Society