Variants of the Busemann–Petty Problem and of the Shephard Problem

Giannopoulos, Apostolos; Koldobsky, Alexander

doi:10.1093/imrn/rnw046

Cited by 6 publications

(12 citation statements)

References 30 publications

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“…Applying (15) and sending δ to zero, we see that the latter inequality in conjunction with (17) implies Proof: Let ε be such that |K ∩ H| ≤ ε |L ∩ H| for all H ∈ Gr n−k , and let D be as in the proof of Theorem 6. By (12), for all H…”

Section: Proof Of Theoremmentioning

confidence: 97%

“…A mixed version of the Busemann-Petty and Shephard problems was posed by Milman and solved in [15]. Namely, if K is a convex body in R n , D is a compact subset of R n and 1 ≤ k ≤ n − 1, then the inequalities |K|H| ≤ |D ∩ H| for all H ∈ Gr n−k imply |K| ≤ |D|.…”

Section: Proof Of Theoremmentioning

confidence: 99%

“…Namely, if K is a convex body in R n , D is a compact subset of R n and 1 ≤ k ≤ n − 1, then the inequalities |K|H| ≤ |D ∩ H| for all H ∈ Gr n−k imply |K| ≤ |D|. Here K|H is the orthogonal projection of K onto H. One can easily modify the proof from [15] to get a slightly stronger version of this result.…”

Section: Proof Of Theoremmentioning

confidence: 99%

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Inequalities for the Radon transform on convex sets

Giannopoulos¹,

Koldobsky²,

Zvavitch³

2021

Preprint

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Several years ago the authors started looking at some problems of convex geometry from a more general point of view, replacing volume by an arbitrary measure. This approach led to new general properties of the Radon transform on convex bodies including an extension of the Busemann-Petty problem and a slicing inequality for arbitrary functions. The latter means that the sup-norm of the Radon transform of any probability density on a convex body of volume one is bounded from below by a positive constant depending only on the dimension. In this note, we prove an inequality that serves as an umbrella for these results. Let K and L be star bodies in R n , let 0 < k < n, and let f, g be non-negative continuous functions on K and L, respectively, so that g

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Section: Proof Of Theoremmentioning

confidence: 97%

Section: Proof Of Theoremmentioning

confidence: 99%

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Inequalities for the Radon transform on convex sets

Giannopoulos¹,

Koldobsky²,

Zvavitch³

2021

Preprint

Self Cite

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“…It is interesting to note that if

K

and

L

are convex bodies in

R^{n}

such that

{normalΠ}_{K} false(t false) ⩽ S_{L} false(t false), for all t ⩾ 0,

then

| K | ⩽ | L | .

This follows from the proof of [, Theorem 1.2].…”

Section: Case Of Double-struckrn N⩾3: Central Hyperplane Sections Anmentioning

confidence: 97%

Distribution functions of sections and projections of convex bodies

Kim

Yaskin

Zvavitch

2016

J. London Math. Soc.

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Typically, when we are given the section (or projection) function of a convex body, it means that in each direction we know the size of the central section (or projection) perpendicular to this direction. Suppose now that we can only get the information about the sizes of sections (or projections), and not about the corresponding directions. In this paper we study to what extent the distribution function of the areas of central sections (or projections) of a convex body can be used to derive some information about the body, its volume, etc

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“…A different kind of volume difference inequality was proved in [14]. If K is any origin-symmetric star body in R n , L is an intersection body, and min ξ∈S n−1 |K ∩ ξ ⊥ | − |L ∩ ξ ⊥ | > 0, where ξ ⊥ is the subspace of R n perpendicular to ξ, then…”

Section: Introductionmentioning

confidence: 99%