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## Studia Mathematica

1999 | 136 | 2 | 147-182
Tytuł artykułu

### Volume ratios in $L_p$-spaces

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
There exists an absolute constant $c_0$ such that for any n-dimensional Banach space E there exists a k-dimensional subspace F ⊂ E with k≤ n/2 such that $inf_{ellipsoid ε ⊂ B_E} (vol(B_E)/vol(ε))^{1/n} ≤ c_0 inf_{zonoid Z ⊂ B_F} (vol(B_F)/vol(Z))^{1/k}$ . The concept of volume ratio with respect to $ℓ_p$-spaces is used to prove the following distance estimate for $2≤ q≤ p < ∞$: $sup_{F ⊂ ℓ_p, dim F=n} inf_{G ⊂ L_q, dim G=n} d(F,G) ∼_{c_{pq}} n^{(q/2)(1/q-1/p)}$.
Słowa kluczowe
Kategorie tematyczne
Czasopismo
Rocznik
Tom
Numer
Strony
147-182
Opis fizyczny
Daty
wydano
1999
otrzymano
1998-09-04
poprawiono
1999-03-05
Twórcy
autor
• Department of Mathematics, Technion, Haifa 32000, Israel
autor
• Math. Seminar Kiel, Ludewig Meyn Str.4, 24098 Kiel, Germany
• Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, IL 61801, U.S.A.
Bibliografia
• [Ba] K. Ball, Normed spaces with a weak-Gordon-Lewis property, in: Functional Analysis (Austin, TX, 1987/1989), Lecture Notes in Math. 1470, Springer, 1991, 36-47.
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• [BF] I. Bárány and Z. Füredi, Computing the volume is difficult, Discrete Comput. Geom. 2 (1987), 319-326.
• [BG] Y. Benyamini and Y. Gordon, Random factorization of operators between Banach spaces, J. Anal. Math. 39 (1981), 45-74.
• [Bo] J. Bourgain, Subspaces of $L_N^∞$, arithmetical diameter and Sidon sets, in; Probability in Banach Spaces V (Medford, 1984), Lecture Notes in Math. 1153, Springer, Berlin, 1985, 96-127.
• [BLM] J. Bourgain, J. Lindenstrauss and V. D. Milman, Approximation of zonoids by zonotopes, Acta Math. 162 (1989), 73-141.
• [BM] J. Bourgain and V. D. Milman, New volume ratio properties for convex symmetric bodies in $ℝ^n$, Invent. Math. 88 (1987), 319-340.
• [BT] J. Bourgain and L. Tzafriri, Embedding $l^k_p$ in subspaces of $L_p$ for p>2, Israel J. Math. 72 (1990), 321-340.
• [Ca] B. Carl, Inequalities of Bernstein-Jackson type and the degree of compactness of operators in Banach spaces, Ann. Inst. Fourier (Grenoble) 35 (1985), no. 3, 79-118.
• [CAR] B. Carl, Inequalities between absolutely (p,q)-summing norms, Studia Math. 69 (1980), 143-148.
• [CP] B. Carl and A. Pajor, Gelfand numbers of operators with values in a Hilbert space, Invent. Math. 94 (1988), 479-504.
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• [DJ] A. Defant and M. Junge, On some matrix inequalities in Banach spaces, Rev. R. Acad. Cien. Exactas Fis. Nat. (Esp.) 0 (1996), 133-140.
• [FJ] T. Figiel and W. B. Johnson, Large subspaces of $l^n_∞$ and estimates of the Gordon-Lewis constant, Israel J. Math. 37 (1980), 92-112.
• [FLM] T. Figiel, J. Lindenstrauss and V. D. Milman, The dimension of almost spherical sections of convex bodies, Acta Math. 139 (1977), 53-94.
• [GE1] S. Geiss, Grothendieck numbers of linear and continuous operators on Banach spaces, Math. Nachr. 148 (1990), 65-79.
• [GTT] E. D. Gluskin, N. Tomczak-Jaegermann and L. Tzafriri, Subspaces of $l^N_p$ of small codimension, Israel J. Math. 79 (1992), 173-192.
• [GO1] Y. Gordon, On the projection and MacPhail constants in $ℓ_p^n$, ibid. 4 (1966), 177-188.
• [GO1] Y. Gordon, Some inequalities for Gaussian processes and applications, ibid. 50 (1985), 177-188.
• [GJ] Y. Gordon and M. Junge, Volume formulas in $L_p$-spaces, Positivity 1 (1997), 7-43.
• [GJ] Y. Gordon and M. Junge, Vector valued Gordon-Lewis property and volume estimates, in preparation.
• [GJN] Y. Gordon, M. Junge and N. J. Nielsen, The relations between volume ratios and new concepts of GL constants, Positivity 1 (1997), 359-379.
• [GMP] Y. Gordon, M. Meyer and A. Pajor, Ratios of volumes and factorization $ℓ_∞$, Illinois J. Math. 40 (1996), 91-107.
• [Gue] O. Guédon, Gaussian version of theorem of Milman and Schechtman, Positivity 1 (1997), 1-5.
• [J] F. John, Extremum problems with inequalities as subsidiary conditions, in: Studies and Essays Presented to R. Courant on his 60th Birthday, Interscience, New York, 1948, 187-204.
• [JU1] M. Junge, Charakterisierung der K-Konvexität durch Volumenquotienten, Master thesis, Kiel, 1989.
• [JU1] M. Junge, Hyperplane conjecture for quotient spaces of $L_p$, Forum Math. 6 (1994), 617-635.
• [L] D. R. Lewis, Finite dimensional subspaces of $L_p$, Studia Math. 63 (1978), 207-212.
• [MaP] M. Marcus and G. Pisier, Random Fourier Series with Application to Harmonic Analysis, Ann. of Math. Stud. 101, Princeton Univ. Press, 1981.
• [MAS] V. Mascioni, On generalized volume ratio numbers, Bull. Sci. Math. 115 (1991), 453-510.
• [M] B. Maurey, Un théorème de prolongement, C. R. Acad. Sci. Paris Sér. A 279 (1974), 329-332.
• [MP] M. Meyer and A. Pajor, Sections of the unit ball of $ℓ_p^n$, J. Funct. Anal. 80 (1988), 109-123.
• [MiP] V. Milman and G. Pisier, Banach spaces with a weak cotype 2 property, Israel J. Math. 54 (1986), 139-158.
• [PT] A. Pajor and N. Tomczak-Jaegermann, Volume ratio and other s-numbers related to local properties of Banach spaces, J. Funct. Anal. 87 (1989), 273-293.
• [PES] 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.
• [PIE] A. Pietsch, Operator Ideals, North-Holland, Amsterdam, 1980.
• [PS1] G. Pisier, Holomorphic semi-groups and the geometry of Banach spaces, Ann. of Math. 115 (1982), 375-392.
• [PS1] G. Pisier, Factorization of Linear Operators and the Geometry of Banach Spaces, CBMS Regional Conf. Ser. in Math. 60, Amer. Math. Soc., 1986, reprinted with corrections, 1987.
• [PS1] G. Pisier, The Volume of Convex Bodies and Banach Space Geometry, Cambridge Tracts in Math. 94, Cambridge Univ. Press, 1989.
• [S] L. Santaló, Un invarianta afin para los cuerpos convexos del espacio de n dimensiones, Portugal. Math. 8 (1949), 155-161.
• [SC] C. Schütt, On the volume of unit balls in Banach spaces, Compositio Math. 47 (1982), 393-407.
• [So] A. Sobczyk, Projections in Minkowski and Banach spaces, Duke J. Math. 8 (1941), 78-106.
• [ST] S. Szarek and M. Talagrand, An "isomorphic" version of the Sauer-Shelah lemma and the Banach-Mazur distance to the cube, in: Geometric Aspects of Functional Analysis (1987-88), Lecture Notes in Math. 1376, Springer, Berlin, 1989, 105-112.
• [TJ] N. Tomczak-Jaegermann, Banach-Mazur Distances and Finite-Dimensional Operator Ideals, Pitman Monographs Surveys Pure Appl. Math. 38, Longman Sci. Tech., Wiley, New York,1989.
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Bibliografia
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