Symmetrization in geometry

Bianchi, Gabriele; Gardner, Richard J.; Gronchi, Paolo

doi:10.1016/j.aim.2016.10.003

Cited by 40 publications

(25 citation statements)

References 32 publications

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“…As in the proof of Theorem 1, we may assume without loss of generality that P = ). For properties of central symmetrization we refer the interested reader to [8]. In particular, recall that the Brunn-Minkowski inequality for intrinsic volumes ( [13]) and (29) yield…”

Section: Proof Of Theoremmentioning

confidence: 99%

On the intrinsic volumes of intersections of congruent balls

Bezdek

2022

Discrete Optimization

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We investigate the intersections of balls of radius r, called r-ball bodies, in Euclidean d-space. An r-lense (resp., r-spindle) is the intersection of two balls of radius r (resp., balls of radius r containing a given pair of points). We prove that among r-ball bodies of given volume, the r-lense (resp., r-spindle) has the smallest inradius (resp., largest circumradius). In general, we upper (resp., lower) bound the intrinsic volumes of r-ball bodies of given inradius (resp., circumradius). This complements and extends some earlier results on volumetric estimates for r-ball bodies. * Keywords and phrases: Euclidean d-space, r-ball body, r-ball polyhedron, intrinsic volume, inradius, circumradius, r-lense, r-spindle.

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Section: Proof Of Theoremmentioning

confidence: 99%

On the intrinsic volumes of intersections of congruent balls

Bezdek

2022

Discrete Optimization

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“…The operator S r (•) corresponds to the so-called "Blaschke-Minkowski" symmetrization, when applied to the support function of a convex body. We refer to [3] and [4] for more information. In view of Lemma 2.2, one naturally expects that there is some sequence of averages of compositions of f with maps from O(n, e n ) that converges in some sense to Sr(f ).…”

Section: Symmetrizationmentioning

confidence: 99%

“…Later on, we will need the fact that the L 1 -norm is preserved under the operator Sr(•) (this is mentioned in [3]) and that if f is in L 2 , then Sr(f ) is also in L 2 . This is done in the following lemma.…”

Section: Symmetrizationmentioning

confidence: 99%

Functions with isotropic sections

Purnaras¹,

Saroglou²

2019

Preprint

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We prove a local version of a recently established theorem by Myroshnychenko, Ryabogin and the second named author. More specifically, we show that if n ≥ 3, g : S n−1 → R is an even bounded measurable function, U is an open subset of S n−1 and the restriction (section) of f onto any great sphere perpendicular to U is isotropic, then C(g)| U = c + a, • and R(g)| U = c ′ , for some fixed constants c, c ′ ∈ R and for some fixed vector a ∈ R n . Here, C(g) denotes the cosine transform and R(g) denotes the Funk transform of g. However, we show that g does not need to be equal to a constant almost everywhere in U ⊥ := u∈U (S n−1 ∩u ⊥ ). For the needs of our proofs, we obtain a new generalization of a result from classical differential geometry, in the setting of convex hypersurfaces, that we believe is of independent interest.

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“…The interest toward the convergence of sequences of successive symmetrizations has risen again in the last years thanks to a series of papers focusing on Steiner symmetrization (for example Klain [13], Bianchi, Burchard, Gronchi and Volcic [2], Bianchi, Klain, Lutwak, Yang and Zhang [5], Volcic [22] and the very recent Asad and Burchard [1]) and on Minkowski symmetrization (as Klartag [14]- [15] and Coupier and Davydov [7]). In Bianchi, Gardner and Gronchi [3]- [4] the authors introduced a wider frame for the study of general symmetrizations, studying the common features and the important properties of these different tools. In particular in [4] they provide a beautiful generalization of Klain's main result in [13] valid for Steiner Symmetrization to many other symmetrizations, including Minkowski's.…”

Section: Introductionmentioning

confidence: 99%

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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Symmetrization in geometry

Cited by 40 publications

References 32 publications

On the intrinsic volumes of intersections of congruent balls

On the intrinsic volumes of intersections of congruent balls

Functions with isotropic sections

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

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