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On asymmetry classes of convex bodies

Part of: General convexity

Published online by Cambridge University Press:  26 February 2010

Rolf Schneider
Rolf Schneider
Affiliation:
Albert-Ludwigs-Universität, Freiburg.
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Asymmetry classes of convex bodies have been introduced and investigated by G. Ewald and G. C. Shephard [2], [3], [6]. These classes are defined as follows. Let denote the set of all convex bodies in n-dimensional Euclidean space ℝn. For K1, K2 ∊ write K1 ∼ K2 if there exist centrally symmetric convex bodies S1, S2 ∊ such that

where + denotes Minkowski addition. Then ∼ is an equivalence relation on and the corresponding classes are called asymmetry classes. The asymmetry class which contains K is denoted by [K].

MSC classification

Secondary: 52A20: Convex sets in $n$ dimensions (including convex hypersurfaces)
Type
Research Article
Information
Mathematika , Volume 21 , Issue 1 , June 1974 , pp. 12 - 18
Copyright
Copyright © University College London 1974

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References

1.Busemann, H.. Convex surfaces (Interscience Publ.: New York, 1958).Google Scholar
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6.Shephard, G. C.. “A pre-Hilbert space consisting of classes of convex sets”, Israel J. Math., 4 (1966), 1–10.CrossRefGoogle Scholar