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## Studia Mathematica

1993 | 106 | 1 | 93-100
Tytuł artykułu

### Balancing vectors and convex bodies

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Treść / Zawartość
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Języki publikacji
EN
Abstrakty
EN
Let U, V be two symmetric convex bodies in $ℝ^n$ and |U|, |V| their n-dimensional volumes. It is proved that there exist vectors $u_1,...,u_n ∈ U$ such that, for each choice of signs $ε_1,...,ε_n = ± 1$, one has $ε_1 u_1 + ... + ε_n u_n ∉ rV$ where $r = (2πe^2)^{-1/2} n^{1/2}(|U|/|V|)^{1/n}$. Hence it is deduced that if a metrizable locally convex space is not nuclear, then it contains a null sequence $(u_n)$ such that the series $∑_{n = 1}^∞ ε_n u_{π(n)}$ is divergent for any choice of signs $ε_n = ± 1$ and any permutation π of indices.
Słowa kluczowe
EN
Kategorie tematyczne
Czasopismo
Rocznik
Tom
Numer
Strony
93-100
Opis fizyczny
Daty
wydano
1993
otrzymano
1992-09-30
poprawiono
1993-02-23
Twórcy
autor
• Institute of Mathematics, Łódź University, 90-238 Łódź, Poland
Bibliografia
• [1] I. K. Babenko, Asymptotic volume of tori and geometry of convex bodies, Mat. Zametki 44 (1988), 177-190 (in Russian).
• [2] K. Ball, Volumes of sections of cubes and related problems, in: Geometric Aspects of Functional Analysis, Israel Seminar (GAFA) 1987-88, Lecture Notes in Math. 1376, Springer, Berlin 1989, 251-260.
• [3] W. Banaszczyk, The Steinitz theorem on rearrangement of series for nuclear spaces, J. Reine Angew. Math. 403 (1990), 187-200.
• [4] W. Banaszczyk, A Beck-Fiala-type theorem for euclidean norms, Europ. J. Combin. 11 (1990), 497-500.
• [5] W. Banaszczyk, Additive Subgroups of Topological Vector Spaces, Lecture Notes in Math. 1466, Springer, Berlin 1991.
• [6] J. Beck and T. Fiala, Integer-making theorems, Discrete Appl. Math. 3 (1981), 1-8.
• [7] J. Beck and J. Spencer, Integral approximation sequences, Math. Programming 30 (1984), 88-98.
• [8] J. Bourgain and S. J. Szarek, The Banach-Mazur distance to the cube and the Dvoretzky-Rogers factorization, Israel J. Math. 62 (1988), 169-180.
• [9] J. W. S. Cassels, An Introduction to the Geometry of Numbers, Springer, Berlin 1959.
• [10] A. Dvoretzky and C. A. Rogers, Absolute and unconditional convergence in normed linear spaces, Proc. Nat. Acad. Sci. U.S.A. 36 (1950), 192-197.
• [11] V. S. Grinberg and S. V. Sevastyanov, Value of the Steinitz constant, Funktsional. Anal. i Prilozhen. 14 (2) (1980), 56-57 (in Russian); English transl.: Functional Anal. Appl. 14 (1980), 125-126.
• [12] B. S. Kashin, On parallelepipeds of minimal volume containing a convex body, Mat. Zametki 45 (1989), 134-135 (in Russian).
• [13] A. Pełczyński and S. J. Szarek, On parallelepipeds of minimal volume containing a convex symmetric body in $ℝ^n$, Math. Proc. Cambridge Philos. Soc. 109 (1991), 125-148.
• [14] S. Sevastyanov, Geometry in the scheduling theory, in: Models and Methods of Optimization, Trudy Inst. Mat. 10, Nauka, Sibirsk. Otdel., Novosibirsk 1988, 226-261 (in Russian).
• [15] J. Spencer, Six standard deviations suffice, Trans. Amer. Math. Soc. 289 (1985), 679-706.
• [16] J. Spencer, Balancing vectors in the max norm, Combinatorica 6 (1986), 55-65.
• [17] J. Spencer, Ten Lectures on the Probabilistic Method, Society for Industrial and Applied Mathematics, Philadelphia, Penn. 1987.
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