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Minimal pairs of convex bodies in two dimensions

Part of: General convexity

Published online by Cambridge University Press:  26 February 2010

Stefan Scholtes
Stefan Scholtes
Affiliation:
Institut für Statistik und mathematische Wirtschaftstheorie, Universität Karlsruhe, 7500 Karlsruhe 1, Germany.
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Extract

In [7] the notion of minimal pairs of convex compact subsets of a Hausdorff topological vector space was introduced and it was conjectured, that minimal pairs in an equivalence class of the Hörmander-Rådström lattice are unique up to translation. We prove this statement for the two-dimensional case. To achieve this we prove a necessary and sufficient condition in terms of mixed volumes that a translate of a convex body in ℝn is contained in another convex body. This generalizes a theorem of Weil (cf. [10]).

MSC classification

Secondary: 52A10: Convex sets in $2$ dimensions (including convex curves)
Type
Research Article
Information
Mathematika , Volume 39 , Issue 2 , December 1992 , pp. 267 - 273
Copyright
Copyright © University College London 1992

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