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A series whose sum range is an arbitrary finite set

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Wojtaszczyk J.
Studia Mathematica
|
2005
|
tom 171
|
nr 3
261-281
EN
In finite-dimensional spaces the sum range of a series has to be an affine subspace. It has long been known that this is not the case in infinite-dimensional Banach spaces. In particular in 1984 M. I. Kadets and K. Woźniakowski obtained an example of a series whose sum range consisted of two points, and asked whether it was possible to obtain more than two, but finitely many points. This paper answers this question affirmatively, by showing how to obtain an arbitrary finite set as the sum range of a series in any infinite-dimensional Banach space.
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A Simpler Proof of the Negative Association Property for Absolute Values of Measures Tied to Generalized Orlicz Balls

100%
Wojtaszczyk J.
Bulletin of the Polish Academy of Sciences. Mathematics
|
2009
|
tom 57
|
nr 1
41-56
EN
Negative association for a family of random variables $(X_i)$ means that for any coordinatewise increasing functions f,g we have $𝔼 (X_{i₁},...,X_{i_k}) g(X_{j₁},...,X_{j_l}) ≤ 𝔼 f(X_{i₁},...,X_{i_k}) 𝔼 g(X_{j₁},...,X_{j_l})$ for any disjoint sets of indices (iₘ), (jₙ). It is a way to indicate the negative correlation in a family of random variables. It was first introduced in 1980s in statistics by Alem & Saxena and Joag-Dev & Proschan, and brought to convex geometry in 2005 by Wojtaszczyk & Pilipczuk to prove the Central Limit Theorem for Orlicz balls. The paper gives a relatively simple proof of negative association of absolute values for a wide class of measures tied to generalized Orlicz balls, including the uniform measures on such balls.
3
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No return to convexity

100%
Wojtaszczyk J.
Studia Mathematica
|
2010
|
tom 199
|
nr 3
227-239
EN
We study the closures of classes of log-concave measures under taking weak limits, linear transformations and tensor products. We investigate which uniform measures on convex bodies can be obtained starting from some class 𝒦. In particular we prove that if one starts from one-dimensional log-concave measures, one obtains no non-trivial uniform mesures on convex bodies.
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