Volume ratios in L p -spaces

Yehoram Gordon; Marius Junge

Studia Mathematica (1999)

  • Volume: 136, Issue: 2, page 147-182
  • ISSN: 0039-3223

Abstract

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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 i n f e l l i p s o i d ε ⊂ B E ( v o l ( B E ) / v o l ( ε ) ) 1 / n ≤ c 0 i n f z o n o i d Z ⊂ B F ( v o l ( B F ) / v o l ( 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 < ∞ : s u p F ⊂ ℓ p , d i m F = n i n f G ⊂ L q , d i m G = n d ( F , G ) ∼ c p q n ( q / 2 ) ( 1 / q - 1 / p ) .

How to cite

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Gordon, Yehoram, and Junge, Marius. "Volume ratios in $L_p$-spaces." Studia Mathematica 136.2 (1999): 147-182. <http://eudml.org/doc/216665>.

@article{Gordon1999,
abstract = {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 $2≤ q≤ p < ∞$: $sup_\{F ⊂ ℓ_p, dim F=n\} inf_\{G ⊂ L_q, dim G=n\} d(F,G) ∼_\{c_\{pq\}\} n^\{(q/2)(1/q-1/p)\}$.},
author = {Gordon, Yehoram, Junge, Marius},
journal = {Studia Mathematica},
keywords = {Carl's inequality; Chevet's inequality; ellipsoid; entropy numbers; Gaussian variables; Gelfand numbers; Gluskin's inequality; volume ratio with respect to ellipsoids; Gordon-Lewis constant; inverse Santaló’s inequality; -convexity; space; space; Marcus-Pisier's theorem; ideal spaces of operators; Gordon-Lewis norms; Meyer-Pajor's inequality; Pajor-Tomczak's inequality; -summing operator; random operator; Santaló’s inequality; volume ratio numbers; zonoid},
language = {eng},
number = {2},
pages = {147-182},
title = {Volume ratios in $L_p$-spaces},
url = {http://eudml.org/doc/216665},
volume = {136},
year = {1999},
}

TY - JOUR
AU - Gordon, Yehoram
AU - Junge, Marius
TI - Volume ratios in $L_p$-spaces
JO - Studia Mathematica
PY - 1999
VL - 136
IS - 2
SP - 147
EP - 182
AB - 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 $2≤ q≤ p < ∞$: $sup_{F ⊂ ℓ_p, dim F=n} inf_{G ⊂ L_q, dim G=n} d(F,G) ∼_{c_{pq}} n^{(q/2)(1/q-1/p)}$.
LA - eng
KW - Carl's inequality; Chevet's inequality; ellipsoid; entropy numbers; Gaussian variables; Gelfand numbers; Gluskin's inequality; volume ratio with respect to ellipsoids; Gordon-Lewis constant; inverse Santaló’s inequality; -convexity; space; space; Marcus-Pisier's theorem; ideal spaces of operators; Gordon-Lewis norms; Meyer-Pajor's inequality; Pajor-Tomczak's inequality; -summing operator; random operator; Santaló’s inequality; volume ratio numbers; zonoid
UR - http://eudml.org/doc/216665
ER -

References

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  1. [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. 
  2. [BaP] K. Ball and A. Pajor, Convex bodies with few faces, Proc. Amer. Math. Soc. 110 (1990), 225-231. Zbl0704.52003
  3. [BF] I. Bárány and Z. Füredi, Computing the volume is difficult, Discrete Comput. Geom. 2 (1987), 319-326. Zbl0628.68041
  4. [BG] Y. Benyamini and Y. Gordon, Random factorization of operators between Banach spaces, J. Anal. Math. 39 (1981), 45-74. Zbl0474.47010
  5. [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. 
  6. [BLM] J. Bourgain, J. Lindenstrauss and V. D. Milman, Approximation of zonoids by zonotopes, Acta Math. 162 (1989), 73-141. Zbl0682.46008
  7. [BM] J. Bourgain and V. D. Milman, New volume ratio properties for convex symmetric bodies in ℝ n , Invent. Math. 88 (1987), 319-340. Zbl0617.52006
  8. [BT] J. Bourgain and L. Tzafriri, Embedding l p k in subspaces of L p for p>2, Israel J. Math. 72 (1990), 321-340. 
  9. [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. Zbl0564.47009
  10. [CAR] B. Carl, Inequalities between absolutely (p,q)-summing norms, Studia Math. 69 (1980), 143-148. Zbl0468.47012
  11. [CP] B. Carl and A. Pajor, Gelfand numbers of operators with values in a Hilbert space, Invent. Math. 94 (1988), 479-504. Zbl0668.47014
  12. [D] R. M. Dudley, The sizes of compact subsets of Hilbert space and continuity of Gaussian processes, J. Funct. Anal. 1 (1967), 290-330. Zbl0188.20502
  13. [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. Zbl0884.47010
  14. [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. Zbl0445.46012
  15. [FLM] T. Figiel, J. Lindenstrauss and V. D. Milman, The dimension of almost spherical sections of convex bodies, Acta Math. 139 (1977), 53-94. Zbl0375.52002
  16. [GE1] S. Geiss, Grothendieck numbers of linear and continuous operators on Banach spaces, Math. Nachr. 148 (1990), 65-79. Zbl0735.47011
  17. [GTT] E. D. Gluskin, N. Tomczak-Jaegermann and L. Tzafriri, Subspaces of l p N of small codimension, Israel J. Math. 79 (1992), 173-192. Zbl0792.46010
  18. [GO1] Y. Gordon, On the projection and MacPhail constants in ℓ p n , ibid. 4 (1966), 177-188. 
  19. [GO1] Y. Gordon, Some inequalities for Gaussian processes and applications, ibid. 50 (1985), 177-188. 
  20. [GJ] Y. Gordon and M. Junge, Volume formulas in L p -spaces, Positivity 1 (1997), 7-43. Zbl0911.46010
  21. [GJ] Y. Gordon and M. Junge, Vector valued Gordon-Lewis property and volume estimates, in preparation. 
  22. [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. Zbl0902.46004
  23. [GMP] Y. Gordon, M. Meyer and A. Pajor, Ratios of volumes and factorization ℓ ∞ , Illinois J. Math. 40 (1996), 91-107. Zbl0843.46008
  24. [Gue] O. Guédon, Gaussian version of theorem of Milman and Schechtman, Positivity 1 (1997), 1-5. Zbl0912.46008
  25. [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. 
  26. [JU1] M. Junge, Charakterisierung der K-Konvexität durch Volumenquotienten, Master thesis, Kiel, 1989. 
  27. [JU1] M. Junge, Hyperplane conjecture for quotient spaces of L p , Forum Math. 6 (1994), 617-635. Zbl0809.52009
  28. [L] D. R. Lewis, Finite dimensional subspaces of L p , Studia Math. 63 (1978), 207-212. Zbl0406.46023
  29. [MaP] M. Marcus and G. Pisier, Random Fourier Series with Application to Harmonic Analysis, Ann. of Math. Stud. 101, Princeton Univ. Press, 1981. 
  30. [MAS] V. Mascioni, On generalized volume ratio numbers, Bull. Sci. Math. 115 (1991), 453-510. Zbl0771.46010
  31. [M] B. Maurey, Un théorème de prolongement, C. R. Acad. Sci. Paris Sér. A 279 (1974), 329-332. 
  32. [MP] M. Meyer and A. Pajor, Sections of the unit ball of ℓ p n , J. Funct. Anal. 80 (1988), 109-123. Zbl0667.46004
  33. [MiP] V. Milman and G. Pisier, Banach spaces with a weak cotype 2 property, Israel J. Math. 54 (1986), 139-158. Zbl0611.46022
  34. [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. Zbl0717.46010
  35. [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. Zbl0718.52007
  36. [PIE] A. Pietsch, Operator Ideals, North-Holland, Amsterdam, 1980. 
  37. [PS1] G. Pisier, Holomorphic semi-groups and the geometry of Banach spaces, Ann. of Math. 115 (1982), 375-392. Zbl0487.46008
  38. [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. 
  39. [PS1] G. Pisier, The Volume of Convex Bodies and Banach Space Geometry, Cambridge Tracts in Math. 94, Cambridge Univ. Press, 1989. 
  40. [S] L. Santaló, Un invarianta afin para los cuerpos convexos del espacio de n dimensiones, Portugal. Math. 8 (1949), 155-161. 
  41. [SC] C. Schütt, On the volume of unit balls in Banach spaces, Compositio Math. 47 (1982), 393-407. 
  42. [So] A. Sobczyk, Projections in Minkowski and Banach spaces, Duke J. Math. 8 (1941), 78-106. Zbl67.0403.03
  43. [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. 
  44. [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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