Advertisement

Periodica Mathematica Hungarica

, Volume 22, Issue 3, pp 183–186 | Cite as

The steinitz lemma inl inf∞ sup2

  • W. Banaszczyk
Article

Abstract

Let (a, b) be a pair of non-negative numbers such that (1)a, b≥1 and (2)a+b≥3. Letu1,...,u n be a sequence of vectors from the set {(x, yR2: |x|, |y|≤1}, withu1+...+u n =0. It is shown that there is a permutation π of indices such that all partial sumsuπ(1)+...+uπ(k) lie in the rectangle |x|≤a, |y|≤b. Conditions (1) and (2) are also necessary.

Mathematics subject classification numbers, (1980)

Primary 52A40 Secondary 05A05 

Key words and phrases

Bounded partial sums rearrangement of vectors Steinitz lemma 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    W.Banaszczyk, The Steinitz constant of the plane,J. reine angew. Math. 373 (1987), 218–220.MR 88e: 52016Google Scholar
  2. [2]
    W.Banaszczyk, The Steinitz theorem on rearrangement of series for nuclear spaces,J. reine angew. Math. 403 (1990), 187–200.Google Scholar
  3. [3]
    I.Bárány, Rearrangement of series in infinite dimensional spaces,Mat. Zametki 46, No. 6. (1989), 10–17 (in russian), translated as Math. Notes.Google Scholar
  4. [4]
    I.Bárány and V. S.Grinberg, A vector-sum theorem in two-dimensional space,Period. Math. Hungar. 16 (1985), 135–138.MR 86j: 52011Google Scholar
  5. [5]
    I.Halperin, Sums of a series, permitting rearrangements,C. R. Math. Rep. Acad. Sci. Canada 8 (1986), 87–102.MR 87m: 40004Google Scholar
  6. [6]
    E.Steinitz, Bedingt konvergente Reihen und konvexe Systeme,J. reine angew. Math. 143 (1913), 128–175.Google Scholar

Copyright information

© Akadémiai Kiadó 1991

Authors and Affiliations

  • W. Banaszczyk
    • 1
  1. 1.Instytut MatematykiUniwersytetu LódzkiegoLódźPoland

Personalised recommendations