On some problems concerning symmetrization operators

Saroglou, Christos

doi:10.1515/forum-2018-0021

Cited by 3 publications

(8 citation statements)

References 13 publications

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“…, n − 1}. When ♦ H = F H , fiber symmetrization with respect to H, equality holds in (35) with B = K n n and f = V n if and only if F H K = K. Hence the corresponding conclusion also holds when ♦ H is monotonic, invariant on H-symmetric sets, and invariant under translations orthogonal to H of H-symmetric sets. In particular, it holds when ♦ H = M H , Minkowski symmetrization with respect to H.…”

Section: Extensions and Variants Of Klain's Theoremmentioning

confidence: 83%

“…for r > 0 and ( 35) is then a consequence of the definition (34) of Ω f . Suppose that f is strictly increasing on B, K ∈ B, and equality holds in (35). Then, in view of (37), we clearly have f (K ∩ rB n ) = f ((♦ H K) ∩ rB n ) for almost all r > 0 and hence, since f is increasing, for all r > 0.…”

Section: Extensions and Variants Of Klain's Theoremmentioning

confidence: 95%

“…Example 5.2. The equality condition in Lemma 5.1 is not enough for our purposes: We would like to have equality in (35) if and only if ♦ H K = K. However, this is not true in general, even when B = K n n . To see this, let H ∈ G(n, i), i ∈ {1, .…”

Section: Extensions and Variants Of Klain's Theoremmentioning

confidence: 99%

“…If f is strictly increasing on B, then equality in (35) implies that ♦ H (K ∩rB n ) = (♦ H K)∩rB n for each r > 0.…”

Section: Extensions and Variants Of Klain's Theoremmentioning

confidence: 99%

“…Proof. Let K ∈ K n , let r > 0, and let x ∈ H. If equality holds in (35) with B = K n n and f = V n , then by Lemma 5.1, we have F H (K ∩ rB n ) = (F H K) ∩ rB n . Using the definition (28) of F H and the Brunn-Minkowski inequality in H ⊥ + x, we obtain…”

Section: Extensions and Variants Of Klain's Theoremmentioning

confidence: 99%

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Convergence of symmetrization processes

Bianchi,

Gardner,

Gronchi

2019

Preprint

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Steiner and Schwarz symmetrizations, and their most important relatives, the Minkowski, Minkowski-Blaschke, fiber, inner rotational, and outer rotational symmetrizations, are investigated. The focus is on the convergence of successive symmetrals with respect to a sequence of i-dimensional subspaces of R n . Such a sequence is called universal for a family of sets if the successive symmetrals of any set in the family converge to a ball with center at the origin. New universal sequences for the main symmetrizations, for all valid dimensions i of the subspaces, are found by combining two groups of new results. In the first, a theorem of Klain for Steiner symmetrization is extended to Schwarz, Minkowski, Minkowski-Blaschke, and fiber symmetrizations, showing that if a sequence of subspaces is drawn from a finite set F of subspaces, the successive symmetrals of any compact convex set converge to a compact convex set that is symmetric with respect to any subspace in F appearing infinitely often in the sequence. The second group of results provides finite sets F of subspaces such that symmetry with respect to each subspace in F implies full rotational symmetry. It is also proved that for Steiner, Schwarz, and Minkowski symmetrizations, a sequence of i-dimensional subspaces is universal for the class of compact sets if and only if it is universal for the class of compact convex sets, and Klain's theorem is shown to hold for Schwarz symmetrization of compact sets.

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Section: Extensions and Variants Of Klain's Theoremmentioning

confidence: 83%

Section: Extensions and Variants Of Klain's Theoremmentioning

confidence: 95%

Section: Extensions and Variants Of Klain's Theoremmentioning

confidence: 99%

“…If f is strictly increasing on B, then equality in (35) implies that ♦ H (K ∩rB n ) = (♦ H K)∩rB n for each r > 0.…”

Section: Extensions and Variants Of Klain's Theoremmentioning

confidence: 99%

Section: Extensions and Variants Of Klain's Theoremmentioning

confidence: 99%

See 3 more Smart Citations

Convergence of symmetrization processes

Bianchi,

Gardner,

Gronchi

2019

Preprint

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Convergence of symmetrization processes

Bianchi¹,

Gardner²,

Gronchi³

2022

Indiana Univ. Math. J.

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

Ulivelli

2021

Can. Math. Bull.

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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).

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On some problems concerning symmetrization operators

Cited by 3 publications

References 13 publications

Convergence of symmetrization processes

Convergence of symmetrization processes

Convergence of symmetrization processes

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

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