Second Derivative Test for Intersection Bodies

Koldobsky, Alexander

doi:10.1006/aima.1998.1719

Cited by 26 publications

(17 citation statements)

References 31 publications

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“…Assume that its radial function ρ L is of class C m (S n−1 ). Let K be the intersection body of L, with radial function ρ K (ϕ) given by (6). Then ρ K (ϕ) is of class C m+ n 2 −1 for 0 < ϕ < π/2, of class C m at ϕ = 0, and of class C m+n−2 at ϕ = π/2.…”

Section: Regularity and Convexity Conditions For An Intersection Bodymentioning

confidence: 99%

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An extension of a result by Lonke to intersection bodies

Alfonseca

2010

Journal of Mathematical Analysis and Applications

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Section: Regularity and Convexity Conditions For An Intersection Bodymentioning

confidence: 99%

“…This function is C ∞ , but L is not equator-convex (see Figure 3). As before, let ρ K be given by (6) with ρ L = ρ L and n = 2n 0 . Then, by Theorem 10, K 2n 0 is an intersection body of a star body, but K 2n 0 +2 is not an intersection body.…”

Section: Partmentioning

confidence: 99%

An extension of a result by Lonke to intersection bodies

Alfonseca

2010

Journal of Mathematical Analysis and Applications

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“…We examine the situation when X ⊕ N Y ∈ I p . There is an earlier result of Koldobsky of this type; see [10], Theorem 4.21 or [5]. Koldobsky shows…”

Section: Introductionmentioning

confidence: 96%

Positive definite distributions and normed spaces

Kalton

Zymonopoulou

2011

Advances in Mathematics

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“…It was proved in [K2,Th. 2] that such spaces with q > 2 satisfy the conditions of Proposition 4 provided that the dimension of X is greater or equal to 3.…”

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confidence: 92%

Positive definite functions and stable random vectors

Koldobsky

2011

Isr. J. Math.

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We say that a random vector X = (X 1 , . . . , Xn) in R n is an n-dimensional version of a random variable Y if, for any a ∈ R n , the random variables a i X i and γ(a)Y are identically distributed, where γ : R n → [0, ∞) is called the standard of X. An old problem is to characterize those functions γ that can appear as the standard of an n-dimensional version. In this paper, we prove the conjecture of Lisitsky that every standard must be the norm of a space that embeds in L 0 . This result is almost optimal, as the norm of any finite-dimensional subspace of Lp with p ∈ (0, 2] is the standard of an n-dimensional version (p-stable random vector) by the classical result of P. Lèvy. An equivalent formulation is that if a function of the form f ( · K ) is positive definite on R n , where K is an origin symmetric star body in R n and f : R → R is an even continuous function, then either the space (R n , · K ) embeds in L 0 or f is a constant function. Combined with known facts about embedding in L 0 , this result leads to several generalizations of the solution of Schoenberg's problem on positive definite functions.

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Second Derivative Test for Intersection Bodies

Cited by 26 publications

References 31 publications

An extension of a result by Lonke to intersection bodies

An extension of a result by Lonke to intersection bodies

Positive definite distributions and normed spaces

Positive definite functions and stable random vectors

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