The geometry of p-convex intersection bodies

Kim, Jaegil; Yaskin, Vladyslav; Zvavitch, Artem

doi:10.1016/j.aim.2011.01.011

Cited by 18 publications

(9 citation statements)

References 26 publications

(11 reference statements)

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“…2 ) ≤ c. Compared with John's classical result, d BM (K, B n 2 ) ≤ √ n for any symmetric convex body K, we see that in many cases the intersection body operation improves convexity in the sense of the Banach-Mazur distance from the ball (see [8] for a similar discussion on quasiconvexity).…”

Section: Introduction and Notationmentioning

confidence: 51%

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On the local convexity of intersection bodies of revolution

Alfonseca,

Kim

2013

Preprint

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One of the fundamental results in Convex Geometry is Busemann's theorem, which states that the intersection body of a symmetric convex body is convex. Thus, it is only natural to ask if there is a quantitative version of Busemann's theorem, i.e., if the intersection body operation actually improves convexity. In this paper we concentrate on the symmetric bodies of revolution to provide several results on the (strict) improvement of convexity under the intersection body operation. It is shown that the intersection body of a symmetric convex body of revolution has the same asymptotic behavior near the equator as the Euclidean ball. We apply this result to show that in sufficiently high dimension the double intersection body of a symmetric convex body of revolution is very close to an ellipsoid in the Banach-Mazur distance. We also prove results on the local convexity at the equator of intersection bodies in the class of star bodies of revolution.

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Section: Introduction and Notationmentioning

confidence: 51%

“…The formula (8), together with ( 13) and (17), gives the modulus of convexity at the equator as follows.…”

Section: Uniform Equatorial Power Type 2 For Intersection Bodiesmentioning

confidence: 99%

On the local convexity of intersection bodies of revolution

Alfonseca,

Kim

2013

Preprint

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“…As a way to measure the "convexity" of a body, we can consider the Banach-Mazur distance from the Euclidean ball. Hensley proved in [6] that the Banach-Mazur distance between the intersection body of any symmetric convex body K and the ball B n 2 is bounded by an absolute constant c > 1, that is, d BM (IK, B n 2 ) ≤ c. Compared with John's classical result, d BM (K, B n 2 ) ≤ √ n for any symmetric convex body K, we see that in many cases the intersection body operation improves convexity in the sense of the Banach-Mazur distance from the ball (see [7] for a similar discussion on quasi-convexity).…”

Section: Introduction and Notationmentioning

confidence: 66%

On the Local Convexity of Intersection Bodies of Revolution

Alfonseca¹,

Kim²

2015

Can. j. math.

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Abstract. One of the fundamental results in Convex Geometry is Busemann's theorem, which states that the intersection body of a symmetric convex body is convex. Thus, it is only natural to ask if there is a quantitative version of Busemann's theorem, i.e., if the intersection body operation actually improves convexity. In this paper we concentrate on the symmetric bodies of revolution to provide several results on the (strict) improvement of convexity under the intersection body operation. It is shown that the intersection body of a symmetric convex body of revolution has the same asymptotic behavior near the equator as the Euclidean ball. We apply this result to show that in sufficiently high dimension the double intersection body of a symmetric convex body of revolution is very close to an ellipsoid in the Banach-Mazur distance. We also prove results on the local convexity at the equator of intersection bodies in the class of star bodies of revolution.

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“…This leads to a randomized version of an extension of the Blaschke-Santaló inequality to the class of convex measures (defined in §2). For this reason we will need the following extension of Busemann's inequality to convex measures from our joint work with D. Cordero-Erausquin and M. Fradelizi [24]; this builds on work by Ball [3], Bobkov [7], Kim, Yaskin and Zvavitch [40]).…”

Section: Dual Settingmentioning

confidence: 99%

Randomized Isoperimetric Inequalities

Paouris

Pivovarov

2017

Convexity and Concentration

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We discuss isoperimetric inequalities for convex sets. These include the classical isoperimetric inequality and that of Brunn-Minkowski, Blaschke-Santaló, Busemann-Petty and their various extensions. We show that many such inequalities admit stronger randomized forms in the following sense: for natural families of associated random convex sets one has stochastic dominance for various functionals such as volume, surface area, mean width and others. By laws of large numbers, these randomized versions recover the classical inequalities. We give an overview of when such stochastic dominance arises and its applications in convex geometry and probability.

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The geometry of p-convex intersection bodies

Cited by 18 publications

References 26 publications

On the local convexity of intersection bodies of revolution

On the local convexity of intersection bodies of revolution

On the Local Convexity of Intersection Bodies of Revolution

Randomized Isoperimetric Inequalities

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