The Geometry of <i>L</i><sub>0</sub>

Kalton, N. J.; Koldobsky, Alexander; Yaskin, Vladyslav; Yaskina, Maryna

doi:10.4153/cjm-2007-044-0

Cited by 25 publications

(30 citation statements)

References 12 publications

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“…On the other hand, as proved in [KKYY,Th. 6.3], Proposition 3: If (R n , · K ) embeds in L 0 , it also embeds in L p for every −n < p < 0.…”

Section: Examplesmentioning

confidence: 80%

“…In particular, it was proved in [KKYY,Th. 6.7] that Proposition 2: Every finite-dimensional subspace of L p , 0 < p ≤ 2 embeds in L 0 .…”

Section: Examplesmentioning

confidence: 92%

“…In particular, the spaces 3 q , q > 2 have this property. In fact, every three-dimensional normed space embeds in L 0 (see [KKYY,Corollary 4.3]). However, starting from dimension 4, there are many normed spaces that do not embed in L 0 .…”

Section: Examplesmentioning

confidence: 98%

“…There are many examples of normed spaces that embed in L 0 , but don't embed in L p , p ∈ (0, 2) (see [KKYY,Th. 6.5]).…”

Section: Examplesmentioning

confidence: 99%

“…We use the following Fourier analytic characterization of embedding in L 0 proved in [KKYY,Th. 3.1]:…”

Section: Proof Of Theoremmentioning

confidence: 99%

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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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“…On the other hand, as proved in [KKYY,Th. 6.3], Proposition 3: If (R n , · K ) embeds in L 0 , it also embeds in L p for every −n < p < 0.…”

Section: Examplesmentioning

confidence: 80%

“…In particular, it was proved in [KKYY,Th. 6.7] that Proposition 2: Every finite-dimensional subspace of L p , 0 < p ≤ 2 embeds in L 0 .…”

Section: Examplesmentioning

confidence: 92%

Section: Examplesmentioning

confidence: 98%

“…There are many examples of normed spaces that embed in L 0 , but don't embed in L p , p ∈ (0, 2) (see [KKYY,Th. 6.5]).…”

Section: Examplesmentioning

confidence: 99%