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Sets homothetic to intersections of their translates

Part of: General convexity

Published online by Cambridge University Press:  26 February 2010

P. McMullen
P. McMullen
Affiliation:
University College London, Gower Street, London WC1E 6BT.
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Abstract

Let S be a compact set in some euclidean space, such that every homo-thetic copy λS of S, with 0 < λ < 1, can be expressed as the intersection of some family of translates of S. It is shown that S has this property precisely when it is star-shaped, and is such that every point in the complement of S is visible from some point (necessarily on the boundary) of the kernel of S. Alternatively, S can be characterized as a compact star-shaped set, whose maximal convex subsets are cap-bodies of its kernel.

MSC classification

Secondary: 52A30: Variants of convex sets (star-shaped, ($m, n$)-convex, etc.)
Type
Research Article
Information
Mathematika , Volume 25 , Issue 2 , December 1978 , pp. 264 - 269
Copyright
Copyright © University College London 1978

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