Isotropic position and inertia ellipsoids and zonoids of the unit ball of a normed n-dimensional space

Milman, Vitali; Pajor, Alain

doi:10.1007/bfb0090049

Cited by 343 publications

(398 citation statements)

References 22 publications

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“…The Busemann-Petty problem has appeared in several places in the literature. See, for example, the articles [B2], [K], [MP,p. 99] and [Be,p.…”

Section: Introductionmentioning

confidence: 99%

Intersection Bodies and the Busemann-Petty Problem

Gardner¹

1994

Transactions of the American Mathematical Society

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Abstract.It is proved that the answer to the Busemann-Petty problem concerning central sections of centrally symmetric convex bodies in ¿-dimensional Euclidean space Ed is negative for a given d if and only if certain centrally symmetric convex bodies exist in Ed which are not intersection bodies. It is also shown that a cylinder in Ed is an intersection body if and only if d < 4 , and that suitably smooth axis-convex bodies of revolution are intersection bodies when d < 4. These results show that the Busemann-Petty problem has a negative answer for d > 5 and a positive answer for d = 3 and d = 4 when the body with smaller sections is a body of revolution.

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“…The Busemann-Petty problem has appeared in several places in the literature. See, for example, the articles [B2], [K], [MP,p. 99] and [Be,p.…”

Section: Introductionmentioning

confidence: 99%

Intersection Bodies and the Busemann-Petty Problem

Gardner¹

1994

Transactions of the American Mathematical Society

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“…Nevertheless, Proposition 4.10 shows that in Löwner's minimal-volume ellipsoid position, we do have R ≤ Cn 1/2−λ /λ where λ depends only on the 2-convexity constant of K. In this case, the concentration result of [ABP03] still holds, with the minor change that L K in (5.1) is replaced by K |x| dx/ √ n (note that this value is always greater than c 1 L K ≥ c 2 , e.g. [MP88]). Although K is no longer isotropic, it is possible to generalize the argument in…”

Section: Gaussian Marginalsmentioning

confidence: 93%

“…It is well known (see, e.g. [MP88]) that for any centrally-symmetric convex K ⊂ R n , there exists a linear transformation such thatK = T (K) is isotropic. Moreover, this map T is unique up to orthogonal transformations.…”

Section: Introductionmentioning

confidence: 99%

Volume distribution in convex bodies

Artstein-Avidan¹,

Giannopoulos²,

Milman³

2015

Mathematical Surveys and Monographs

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Abstract. We consider convex sets whose modulus of convexity is uniformly quadratic. First, we observe several interesting relations between different positions of such "2-convex" bodies; in particular, the isotropic position is a finite volume-ratio position for these bodies. Second, we prove that high dimensional 2-convex bodies posses one-dimensional marginals that are approximately Gaussian. Third, we improve for 1 < p ≤ 2 some bounds on the isotropic constant of quotients of subspaces of L p and S m p , the Schatten Class space.

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“…D We suspect that c could be 1. For related problems, see [30]. From Theorem 2.13, we have the following = fy-i-m'fy-MiK).…”

Section: Characterizations and Inequalities Of Intersection Bodiesmentioning

confidence: 99%

Centered bodies and dual mixed volumes

Zhang¹

1994

Trans. Amer. Math. Soc.

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Abstract. We establish a number of characterizations and inequalities for intersection bodies, polar projection bodies and curvature images of projection bodies in R" by using dual mixed volumes. One of the inequalities is between the dual Quermassintegrals of centered bodies and the dual Quermassintegrals of their central (n -l)-slices. It implies Lutwak's affirmative answer to the Busemann-Petty problem when the body with the smaller sections is an intersection body. We introduce and study the intersection body of order i of a star body, which is dual to the projection body of order i of a convex body. We show that every sufficiently smooth centered body is a generalized intersection body. We discuss a type of selfadjoint elliptic differential operator associated with a convex body. These operators give the openness property of the class of curvature functions of convex bodies. They also give an existence theorem related to a well-known uniqueness theorem about deformations of convex hypersurfaces in global differential geometry.

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Isotropic position and inertia ellipsoids and zonoids of the unit ball of a normed n-dimensional space

Cited by 343 publications

References 22 publications

Intersection Bodies and the Busemann-Petty Problem

Intersection Bodies and the Busemann-Petty Problem

Volume distribution in convex bodies

Centered bodies and dual mixed volumes

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