# On some vector balancing problems

Studia Mathematica (1997)

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

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topGiannopoulos, 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

top- [1] K. Ball and A. Pajor, Convex bodies with few faces, Proc. Amer. Math. Soc. 110 (1990), 225-231.
- [2] W. Banaszczyk, Balancing vectors and convex bodies, Studia Math. 106 (1993), 93-100.
- [3] W. Banaszczyk, A Beck-Fiala-type theorem for Euclidean norms, European J. Combin. 11 (1990), 497-500.
- [4] W. Banaszczyk and S. J. Szarek, Lattice coverings and Gaussian measures of n-dimensional convex bodies, Discrete Comput. Geom., to appear.
- [5] I. Bárány and V. S. Grinberg, On some combinatorial questions in finite-dimensional spaces, Linear Algebra Appl. 41 (1981), 1-9.
- [6] J. Beck and T. Fiala, Integer-making theorems, Discrete Appl. Math. 3 (1981), 1-8.
- [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.
- [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] D. Kleitman, On a combinatorial conjecture of Erdős, J. Combin. Theory Ser. A 1 (1966), 209-214.
- [10] Z. Sidák, On multivariate normal probabilities of rectangles: their dependence on correlation, Ann. Math. Statist. 39 (1968), 1425-1434.
- [11] J. Spencer, Six standard deviations suffice, Trans. Amer. Math. Soc. 289 (1985), 679-706.
- [12] J. Spencer, Balancing vectors in the max norm, Combinatorica 6 (1) (1986), 55-65.
- [13] J. Spencer, Ten Lectures on the Probabilistic Method, SIAM, Philadelphia, Penn., 1994.
- [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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