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Generalization of Klain’s theorem to Minkowski symmetrization of compact sets and related topics

Part of: General convexity Metric geometry

Published online by Cambridge University Press:  02 November 2021

Jacopo Ulivelli Open the ORCID record for Jacopo Ulivelli [Opens in a new window]
Jacopo Ulivelli*
Affiliation:
Department of Mathematics, “G. Castelnuovo” Sapienza University of Rome, Piazzale Aldo Moro, 5, Rome 00185, Italy
*
e-mail: ulivelli@mat.uniroma1.it
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Abstract

We shall prove a convergence result relative to sequences of Minkowski symmetrals of general compact sets. In particular, we investigate the case when this process is induced by sequences of subspaces whose elements belong to a finite family, following the path marked by Klain in Klain (2012, Advances in Applied Mathematics 48, 340–353), and the generalizations in Bianchi et al. (2019, Convergence of symmetrization processes, preprint) and Bianchi et al. (2012, Indiana University Mathematics Journal 61, 1695–1710). We prove an analogous result for fiber symmetrization of a specific class of compact sets. The idempotency for symmetrizations of this family of sets is investigated, leading to a simple generalization of a result from Klartag (2004, Geometric and Functional Analysis 14, 1322–1338) regarding the approximation of a ball through a finite number of symmetrizations, and generalizing an approximation result in Fradelizi, Làngi and Zvavitch (2020, Volume of the Minkowski sums of star-shaped sets, preprint).

Keywords

Convex bodycompact setSteiner symmetrizationSchwarz symmetrizationMinkowski symmetrizationfiber symmetrizationreflectionMinkowski additionidempotency

MSC classification

Primary: 52A20: Convex sets in $n$ dimensions (including convex hypersurfaces) 52A38: Length, area, volume
Secondary: 52A30: Variants of convex sets (star-shaped, ($m, n$)-convex, etc.) 51F15: Reflection groups, reflection geometries 52A39: Mixed volumes and related topics
Type
Article
Information
Canadian Mathematical Bulletin , Volume 66 , Issue 1 , March 2023 , pp. 124 - 141
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Copyright
© Canadian Mathematical Society, 2021

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