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Generalizing the Johnson-Lindenstrauss lemma to k-dimensional affine subspaces

Alon Dmitriyuk, Yehoram Gordon (2009)

Studia Mathematica

Let ε > 0 and 1 ≤ k ≤ n and let W l l = 1 p be affine subspaces of ℝⁿ, each of dimension at most k. Let m = O ( ε - 2 ( k + l o g p ) ) if ε < 1, and m = O(k + log p/log(1 + ε)) if ε ≥ 1. We prove that there is a linear map H : m such that for all 1 ≤ l ≤ p and x , y W l we have ||x-y||₂ ≤ ||H(x)-H(y)||₂ ≤ (1+ε)||x-y||₂, i.e. the distance distortion is at most 1 + ε. The estimate on m is tight in terms of k and p whenever ε < 1, and is tight on ε,k,p whenever ε ≥ 1. We extend these results to embeddings into general normed spaces Y.

Józef Marcinkiewicz (1910-1940) - on the centenary of his birth

Lech Maligranda (2011)

Banach Center Publications

Józef Marcinkiewicz’s (1910-1940) name is not known by many people, except maybe a small group of mathematicians, although his influence on the analysis and probability theory of the twentieth century was enormous. This survey of his life and work is in honour of the 100 t h anniversary of his birth and 70 t h anniversary of his death. The discussion is divided into two periods of Marcinkiewicz’s life. First, 1910-1933, that is, from his birth to his graduation from the University of Stefan Batory in Vilnius,...

Khinchin inequality and Banach-Saks type properties in rearrangement-invariant spaces

F. A. Sukochev, D. Zanin (2009)

Studia Mathematica

We study the class of all rearrangement-invariant ( = r.i.) function spaces E on [0,1] such that there exists 0 < q < 1 for which k = 1 n ξ k E C n q , where ξ k k 1 E is an arbitrary sequence of independent identically distributed symmetric random variables on [0,1] and C > 0 does not depend on n. We completely characterize all Lorentz spaces having this property and complement classical results of Rodin and Semenov for Orlicz spaces e x p ( L p ) , p ≥ 1. We further apply our results to the study of Banach-Saks index sets in...

Markov convexity and local rigidity of distorted metrics

Manor Mendel, Assaf Naor (2013)

Journal of the European Mathematical Society

It is shown that a Banach space admits an equivalent norm whose modulus of uniform convexity has power-type p if and only if it is Markov p -convex. Counterexamples are constructed to natural questions related to isomorphic uniform convexity of metric spaces, showing in particular that tree metrics fail to have the dichotomy property.

On a discrete version of the antipodal theorem

Krzysztof Oleszkiewicz (1996)

Fundamenta Mathematicae

The classical theorem of Borsuk and Ulam [2] says that for any continuous mapping f : S k k there exists a point x S k such that f(-x) = f(x). In this note a discrete version of the antipodal theorem is proved in which S k is replaced by the set of vertices of a high-dimensional cube equipped with Hamming’s metric. In place of equality we obtain some optimal estimates of i n f x | | f ( x ) - f ( - x ) | | which were previously known (as far as the author knows) only for f linear (cf. [1]).

On multivalued martingales, multimeasures and multivalued Radon-Nikodym property

Mohamed Zohry (2004)

Bollettino dell'Unione Matematica Italiana

In this paper we prove a representation result for essentially bounded multivalued martingales with nonempty closed convex and bounded values in a real separable Banach space. Then we turn our attention to the interplay between multimeasures and multivalued Riesz representations. Finally, we give the multivalued Radon-Nikodym property.

On the best constant in the Khinchin-Kahane inequality

Rafał Latała, Krzysztof Oleszkiewicz (1994)

Studia Mathematica

We prove that if r i is the Rademacher system of functions then ( ʃ i = 1 n x i r i ( t ) 2 d t ) 1 / 2 2 ʃ i = 1 n x i r i ( t ) d t for any sequence of vectors x i in any normed linear space F.

On the convergence of certain sums of independent random elements

Juan Carlos Ferrando (2002)

Commentationes Mathematicae Universitatis Carolinae

In this note we investigate the relationship between the convergence of the sequence { S n } of sums of independent random elements of the form S n = i = 1 n ε i x i (where ε i takes the values ± 1 with the same probability and x i belongs to a real Banach space X for each i ) and the existence of certain weakly unconditionally Cauchy subseries of n = 1 x n .

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