Geodesic
Generalization of Klain’s theorem to Minkowski symmetrization of compact sets and related topics
Canadian mathematical bulletin, Tome 66 (2023) no. 1, pp. 124-141

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DOI

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).
DOI : 10.4153/S0008439521000904
Mots-clés : Convex body, compact set, Steiner symmetrization, Schwarz symmetrization, Minkowski symmetrization, fiber symmetrization, reflection, Minkowski addition, idempotency 
Ulivelli, Jacopo. Generalization of Klain’s theorem to Minkowski symmetrization of compact sets and related topics. Canadian mathematical bulletin, Tome 66 (2023) no. 1, pp. 124-141. doi: 10.4153/S0008439521000904
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     title = {Generalization of {Klain{\textquoteright}s} theorem to {Minkowski} symmetrization of compact sets and related topics},
     journal = {Canadian mathematical bulletin},
     pages = {124--141},
     year = {2023},
     volume = {66},
     number = {1},
     doi = {10.4153/S0008439521000904},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/S0008439521000904/}
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