On some vector balancing problems

Apostolos Giannopoulos

Studia Mathematica (1997)

  • Volume: 122, Issue: 3, page 225-234
  • ISSN: 0039-3223

Abstract

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Let V be an origin-symmetric convex body in n , n≥ 2, of Gaussian measure γ n ( V ) 1 / 2 . It is proved that for every choice u 1 , . . . , u n of vectors in the Euclidean unit ball B n , there exist signs ε j - 1 , 1 with ε 1 u 1 + . . . + ε n u n ( c l o g n ) V . The method used can be modified to give simple proofs of several related results of J. Spencer and E. D. Gluskin.

How to cite

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Giannopoulos, Apostolos. "On some vector balancing problems." Studia Mathematica 122.3 (1997): 225-234. <http://eudml.org/doc/216373>.

@article{Giannopoulos1997,
abstract = {Let V be an origin-symmetric convex body in $ℝ^n$, n≥ 2, of Gaussian measure $γ_n(V)≥ 1/2$. It is proved that for every choice $u_1,...,u_n$ of vectors in the Euclidean unit ball $B_n$, there exist signs $ε_j ∈ \{-1,1\}$ with $ε_\{1\}u_\{1\} + ... + ε_\{n\}u_\{n\} ∈ (clogn)V$. The method used can be modified to give simple proofs of several related results of J. Spencer and E. D. Gluskin.},
author = {Giannopoulos, Apostolos},
journal = {Studia Mathematica},
keywords = {balancing problem},
language = {eng},
number = {3},
pages = {225-234},
title = {On some vector balancing problems},
url = {http://eudml.org/doc/216373},
volume = {122},
year = {1997},
}

TY - JOUR
AU - Giannopoulos, Apostolos
TI - On some vector balancing problems
JO - Studia Mathematica
PY - 1997
VL - 122
IS - 3
SP - 225
EP - 234
AB - Let V be an origin-symmetric convex body in $ℝ^n$, n≥ 2, of Gaussian measure $γ_n(V)≥ 1/2$. It is proved that for every choice $u_1,...,u_n$ of vectors in the Euclidean unit ball $B_n$, there exist signs $ε_j ∈ {-1,1}$ with $ε_{1}u_{1} + ... + ε_{n}u_{n} ∈ (clogn)V$. The method used can be modified to give simple proofs of several related results of J. Spencer and E. D. Gluskin.
LA - eng
KW - balancing problem
UR - http://eudml.org/doc/216373
ER -

References

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  1. [1] K. Ball and A. Pajor, Convex bodies with few faces, Proc. Amer. Math. Soc. 110 (1990), 225-231. Zbl0704.52003
  2. [2] W. Banaszczyk, Balancing vectors and convex bodies, Studia Math. 106 (1993), 93-100. Zbl0810.46009
  3. [3] W. Banaszczyk, A Beck-Fiala-type theorem for Euclidean norms, European J. Combin. 11 (1990), 497-500. Zbl0736.51011
  4. [4] W. Banaszczyk and S. J. Szarek, Lattice coverings and Gaussian measures of n-dimensional convex bodies, Discrete Comput. Geom., to appear. Zbl0872.52007
  5. [5] I. Bárány and V. S. Grinberg, On some combinatorial questions in finite-dimensional spaces, Linear Algebra Appl. 41 (1981), 1-9. Zbl0467.90079
  6. [6] J. Beck and T. Fiala, Integer-making theorems, Discrete Appl. Math. 3 (1981), 1-8. Zbl0473.05046
  7. [7] E. D. Gluskin, Extremal properties of orthogonal parallelepipeds and their applications to the geometry of Banach spaces, Math. USSR-Sb. 64 (1989), 85-96. Zbl0668.52002
  8. [8] B. S. Kashin, An analogue of Menshov's "correction" theorem for discrete orthonormal systems, Mat. Zametki 46 (6) (1989), 67-74 (in Russian); English transl.: Math. Notes 46 (5-6) (1989), 934-939. 
  9. [9] D. Kleitman, On a combinatorial conjecture of Erdős, J. Combin. Theory Ser. A 1 (1966), 209-214. Zbl0148.01105
  10. [10] Z. Sidák, On multivariate normal probabilities of rectangles: their dependence on correlation, Ann. Math. Statist. 39 (1968), 1425-1434. Zbl0169.50102
  11. [11] J. Spencer, Six standard deviations suffice, Trans. Amer. Math. Soc. 289 (1985), 679-706. Zbl0577.05018
  12. [12] J. Spencer, Balancing vectors in the max norm, Combinatorica 6 (1) (1986), 55-65. Zbl0593.90110
  13. [13] J. Spencer, Ten Lectures on the Probabilistic Method, SIAM, Philadelphia, Penn., 1994. Zbl0822.05060
  14. [14] S. J. Szarek and E. Werner, Confidence regions for means of multivariate normal distributions and a non-symmetric correlation inequality for Gaussian measure, preprint. 

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